Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Refinements of Some Classical Inequalities Involving Sinc and Hyperbolic Sinc Functions

Refinements of Some Classical Inequalities Involving Sinc and Hyperbolic Sinc Functions Annales Mathematicae Silesianae 37 (2023), no. 1, 1–15 DOI: 10.2478/amsil-2022-0019 REFINEMENTS OF SOME CLASSICAL INEQUALITIES INVOLVING SINC AND HYPERBOLIC SINC FUNCTIONS Yogesh J. Bagul, Sumedh B. Thool , Christophe Chesneau, Ramkrishna M. Dhaigude Abstract. Several bounds of trigonometric-exponential and hyperbolic-expo- nential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions. 1. Introduction Consider the sinc function defined by sinc x = (sin x)=x; for x 6= 0 and sinc x = 1; for x = 0: A hyperbolic sinc function is defined similarly. Let us now cite some inequalities for sinc and hyperbolic sinc functions pertaining to the main results of this paper. First, the classical inequalities x sin x 2 2 (1.1) cos < < cos arccos  x ; 0 < x < ; x   2 were established by K.S.K. Iyengar, B.S. Madhava Rao and T.S. Nanjundiah in a little-known paper [9]. See also [14]. Recently, J. Sándor ([18]) offered Received: 25.03.2022. Accepted: 02.11.2022. Published online: 23.11.2022. (2020) Mathematics Subject Classification: 26D05, 26D07, 33B10. Key words and phrases: trigonometric-exponential, hyperbolic-exponential, Mitrinović- Adamović inequality, Lazarević inequality, Iyengar-Madhava Rao-Nanjudiah inequality. c 2022 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). 2 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude a new proof to the left inequality of (1.1) and proved its hyperbolic counterpart as follows: x sinh x (1.2) cosh p < ; x > 0: R. Klén, M. Visuri, and M. Vuorinen ([10]) found the following inequalities h  i h  i 2 3 x sin x x (1.3) cos < < cos ; 0 < x < : 2 x 3 2 Y. Lv, G. Wang, and Y. Chu ([11]) obtained the following: h  i h  i 4=3 a x sin x x (1.4) cos < < cos ; 0 < x < ; 2 x 2 2 where a = (ln(=2))=(ln 2)  1:30299: The left inequality of (1.4) is sharper than the corresponding left inequality of (1.3), whereas the right inequality of (1.3) is better than that of (1.4). The analogous inequality to (1.4) is the following one: h  i 4=3 x sinh x (1.5) cosh < < cosh x; x > 0; 2 x which can be seen in [15,16,20]. Exponential-type bounds for sinc and hyper- bolic sinc functions were obtained by Chesneau and Bagul in [6]. They are given below. We have 2 2 sin x x x =6 (1.6) e < < e ; 0 < x < ; x 2 where = 4 ln(2=)= , and 2 2 sinh x x x =6 (1.7) e < < e ; 0 < x < r; where r > 0 and  = ln[(sinh r)=r]=r : For different refinements, generalizations, and recent developments regard- ing inequalities involving the sinc and hyperbolic sinc functions, we refer the reader to [2–5,7,12,13,17,21–26]. This article aims to present new generalized bounds for sinc and hyperbolic sinc functions. Our bounds are trigonometric- exponential and hyperbolic-exponential in nature and they refine some exist- ing bounds in the literature. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 3 We consider the following plan: Section 2 presents some preliminaries and lemmas. The main results are given in Section 3. Section 4 ends the article with some particular cases and discussions. 2. Preliminaries and lemmas The following power series expansions involving Bernoulli numbers can be found in [8, 1.411]: 2n 2n 2 (2 1) 2n1 (2.1) tan x = jB jx ; jxj < ; 2n (2n)! 2 n=1 2n 1 2 2n1 (2.2) cot x = jB jx ; jxj < ; 2n x (2n)! n=1 and 2n 2n 2 (2 1) 2n1 (2.3) tanh x = B x ; jxj < ; 2n (2n)! 2 n=1 where B are the even indexed Bernoulli numbers. 2n We will also use the following l’Hôpital’s rule of monotonicity. Lemma 1 (l’Hôpital’s rule of monotonicity [1]). Let f; g : [a; b] ! R be two continuous functions which are differentiable on (a; b) and g 6= 0 on (a; b): 0 0 If f =g is increasing (or decreasing) on (a; b); then the functions (f(x) f(a))=(g(x) g(a)) and (f(x) f(b))=(g(x) g(b)) are also increasing (or 0 0 decreasing) on (a; b): If f =g is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 2 ([2, Lemma 4]). For x > 0; the function sinh x x cosh x k(x) = x sinh x is strictly increasing. Additionally, we prove the following auxiliary results which can be of in- dependent interest. 4 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Lemma 3 (Sharp upper bound for hyperbolic sinc). For x 6= 0; it is true that sinh x (x + sinh x)(1 + cosh x) (2.4) < : x 4x Proof. Due to the symmetry of the functions involved at both sides, it suffices to prove (2.4) for x > 0: We first consider f(x) = x + x sinh x 4 cosh x + 4: Then f (x) = 2x + x cosh x 3 sinh x > 0; due to well-known Cusa–Huygens inequality ([2]). Therefore, f(x) is increasing for x > 0 and we get f(x) > f(0), i.e., x + x sinh x 4(cosh x 1) > 0; which can be written as 4(cosh x 1)(cosh x + 1) < x(x + sinh x)(1 + cosh x) or 4 sinh x < x(x + sinh x)(1 + cosh x): This gives the required inequality (2.4). Remark 1. For x 6= 0; it is not difficult to prove (x + sinh x)(1 + cosh x) 2 + cosh x < : 4x 3 Thus, we have sinh x (x + sinh x)(1 + cosh x) 2 + cosh x (2.5) < < ; x 6= 0: x 4x 3 Remark 2. A double inequality analogous to (2.5) also holds in the case of trigonometric functions. It is stated as sin x (x + sin x)(1 + cos x) 2 + cos x (2.6) < < ; x 6= 0: x 4x 3 We skip the proof of (2.6) because it is very similar to that of (2.5). Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 5 Lemma 4 (Refined lower bound for Wilker-type inequality). For x 6= 0; it is true that x x x(cosh x 1)(sinh x x) (2.7) + > 2 + sinh x tanh x 2 sinh x x(sinh x x) = 2 + > 2: 2(1 + cosh x) Proof. It is enough to prove that x x x(cosh x 1)(sinh x x) + > 2 + : sinh x tanh x 2 sinh x Equivalently, it corresponds to 2x + 2x sinh x cosh x > 4 sinh x + x(cosh x 1)(sinh x x) or 2 2 x + x sinh x cosh x 4 sinh x + x sinh x + x cosh x > 0; i.e., x(x + sinh x)(1 + cosh x) > 4 sinh x; which is true by Lemma 3. Remark 3. The inequality (2.7) is a refinement of the Wilker-type in- equality for hyperbolic functions established by Wu and Debnath ([19]). Remark 4. It is interesting to see that the circular counterpart of (2.7) is also true for all non-zero real numbers. It is stated as follows: x x x(cos x 1)(sin x x) (2.8) + > 2 + sin x tan x 2 sin x x(x sin x) = 2 + > 2; x 6= 0: 2(1 + cos x) The proof of (2.8) is quite similar to that of (2.7). The importance of (2.7) lies in the fact that it is the sharpest Wilker-type inequality of its kind so far in the literature, and it holds for all non-zero real numbers, although its sharpness can be observed in (0; ) and (; 0) only. 6 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Lemma 5. For x 6= 0; it is true that x x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] + > 2 + sinh x tanh x p sinh (2x=p) if p  2: Proof. Let g(x) = 2 + x tanh x (x sech x) : Differentiation yields 0 2 2 2 g (x) = tanh x + 2x sech x tanh x x sech x x sinh x 1 2 2 = + 2x sech x tanh x > 0: cosh x x cosh x Hence, g(x) is increasing and we have that g(x=2)  g(x=p) if x=2  x=p, i.e., p  2: Now, we have x x x x x g = 2 + tanh sech 2 2 2 2 2 x x sinh(x=2) = 2 + 2 cosh(x=2) 4 cosh (x=2) x x sinh x = 2 + 2(1 + cosh x) 2(1 + cosh x) x(sinh x x) = 2 + 2(1 + cosh x) x(cosh x 1)(sinh x x) = 2 + : 2 sinh x Similarly, we establish that x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] g = 2 + : p sinh (2x=p) By making use of Lemma 4, the conclusion of Lemma 5 follows.  Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 7 3. Main results We are now in a position to state and prove our main results. Theorem 1. For p > 1; we define  : (0; =2] ! R by h i sin x ln x cos(x=p) (x) = : Then 1.  is strictly increasing if p  3; 2.  is strictly decreasing if p  2: Proof. We write ln [(sin x)=x] ln[cos(x=p)] ( ) (x) 1 p (x) = = ; x  (x) where ( ) (x) = ln [(sin x)=x]ln[cos(x=p)] and  (x) = x , with ( ) (0+) = 1 p 2 1 p 0 =  (0): By differentiation, we get ( ) (x) 1 x cos x sin x tan(x=p) = p + (x) 2p x sin x x 1 tan(x=p) cot x p = + p : 2p x x x Utilizing (2.1) and (2.2), we obtain " # 1 1 2n 2n 2n X X ( ) (x) 1 2 (2 1) p 2 2n2 2n2 = jB jx jB jx 2n 2n 0 2n1 (x) 2p p  (2n)! (2n)! n=1 n=1 2n 2n 1 2 2 1 2n2 = p jB jx : 2n 2n1 2p (2n)! p n=1 2n 2n1 By Lemma 1,  will be strictly increasing if (2 1)=p p > 0, i.e., 2n 2n 2n 1=2n 2 1 > p or p < h(n) := (2 1) : And it is easy to show that h(n) is strictly increasing for n = 1; 2; : This implies that p  inffh(n) : n = 1; 2;g = h(1) = 3: Similarly, we can say that  will be strictly 2n 2n1 decreasing if we have (2 1)=p p < 0; or p > h(n): So, we get p  supfh(n) : n = 1; 2;g = lim h(n) = 2: This completes the proof n!1 of Theorem 1.  8 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Next, by l’Hôpital’s rule,  (0+) = lim  (x) = 1=(2p ) 1=6; and p x!0 p (=2) = (4= ) ln [2=[ cos(=(2p))]] : Hence, we immediately deduce the following corollaries: Corollary 1. If 1 < p  3 and 0 < x  =2; then the best possible constants and such that the inequalities 1 1 x 2 sin x x 2 x x 1 1 cos e < < cos e p x p 2 2 hold are 1=(2p ) 1=6 and (4= ) ln [2=[ cos(=(2p))]], respectively. Corollary 2. If p  2 and 0 < x  =2; then the inequalities 2 2 x sin x x x x 1 1 cos e < < cos e p x p hold with the best possible constants and which are as defined in the 1 1 Corollary 1. An analogous result involving hyperbolic functions is formulated in the following theorem. Theorem 2. For p > 0 and r > 0; we define a function : (0; r) ! R by h i sinh x ln x cosh(x=p) (x) = : Then is strictly decreasing if p  2: In particular, if p  2; then the best possible constants and such that the inequalities 2 2 x 2 sinh x x 2 x x 2 2 (3.1) cosh e < < cosh e ; 0 < x < r; p x p 2 2 hold are ln ((sinh r)=[r cosh(r=p)]) =r and 1=6 1=(2p ), respectively. Proof. Set ( ) (x) = ln [(sinh x)=x] ln [cosh(x=p)] and (x) = x : 1 p 2 Clearly ( ) (0+) = 0 = (0) and (x) = ( ) (x)= (x): In view of using 1 p 2 p 1 p 2 Lemma 1, we differentiate and obtain ( ) (x) 1 coth x p tanh(x=p) 1 = p := ( ) (x): 3 p (x) 2p x x x 2p 2 Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 9 Then, we get p p 2p 1 x 1 x 0 2 2 ( ) (x) = cosech x coth x + sech + tanh 2 3 2 x x x px p x p p x x x x x x = + 2 + sech tanh x sinh x tanh x p p p p p x x = + 2 x sinh x tanh x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] p sinh (2x=p) By Lemma 1 and Lemma 5, we conclude that is strictly decreasing if p  2: Consequently, (0+) > (x) > (r); 0 < x < r: p p p The desired inequalities (3.1) follow due to the limits (0+) = 1=6 1=(2p ) and (r) = ln [(sinh r)=[r cosh(r=p)]] =r : Theorem 3. For p > 1; we define ' : (0; =2] ! R by h i sin x ln x cosh(x=p) ' (x) = : Then ' is strictly decreasing if p  2: In particular, if p  2; then the best possible constants and such that the inequalities 3 3 x 2 sin x x 2 x x 3 3 (3.2) cosh e < < cosh e ; 0 < x p x p 2 2 2 hold are (4= ) ln [2=[ cosh(=(2p))]] and 1=(2p ) + 1=6 , respectively. Proof. We begin with ln (sin x=x) ln(cosh x=p) (' ) (x) 1 p ' (x) = = ; x ' (x) 2 10 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude where (' ) (x) = ln [(sin x)=x]ln[cosh(x=p)] and ' (x) = x with (' ) (0+) = 1 p 2 1 p 0 = ' (0): Differentiation gives (' ) (x) 1 x cos x sin x tanh(x=p) = p ' (x) 2p x sin x x 1 cot x p tanh(x=p) = p : 2p x x x Utilizing (2.1) and (2.3), we obtain " # 1 1 2n 2n 2n X X (' ) (x) 1 p 2 2 (2 1) 2n2 2n2 = jB jx B x 2n 2n 2n1 ' (x) 2p (2n)! p  (2n)! n=1 n=1 2n 2n 1 2 (2 1) 2n2 = pjB j B x 2n 2n 2n1 2p (2n)! p n=1 2n 1 2 2n2 := a x ; 2p (2n)! n=1 2n 2n1 where a = pjB j [(2 1)=p ]B : By Lemma 1, ' will be strictly n 2n 2n p decreasing if a > 0: But, a is always positive for B < 0 irrespective of n n 2n p: So we consider the case when B > 0: In this case, a > 0 implies that 2n n 2n 2n 2n 2n 2n 1=(2n) jB j > [(2 1)=p ]B or p > (2 1), i.e., p > (2 1) := h(n) 2n 2n and h(n) being strictly increasing, we write p  supfh(n) : n = 1; 2;g = lim h(n) = 2: Finally, ' (0+) > ' (x) > ' (=2), and the limits n!1 p p p 2 2 ' (0+) = 1=(2p ) + 1=6 and ' (=2) = (4= ) ln [2=[ cosh(=(2p))]] p p give the inequalities (3.2). Theorem 4. For p  1 and r < p=2; we define  : (0; r) ! R by h i ln sinh x cos(x=p) (x) = : Then  is strictly increasing. In particular, if p  1; then the best possible constants and such that the inequalities 4 4 2 2 x x 4 4 e sinh x e (3.3) < < ; 0 < x < r cos(x=p) x cos(x=p) 2 2 hold are ln [r=[(sinh r)(cos(r=p))]] =r and 1=6 1=(2p ), respectively. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 11 Proof. We have ( ) (x) ( ) (0+) 1 p 1 p (x) = ; (x)  (0) 2 2 where ( ) (x) = ln (x=sinh x)ln [cos(x=p)] and  (x) = x with ( ) (0+) = 1 p 2 1 p 0 =  (0): Differentiation yields ( ) (x) 1 sinh x x cosh x 1 tan(x=p) = + ; 2 2 (x) 2 x sinh x 2p (x=p) which is strictly increasing because of Lemma 2 and the fact that (tan x)=x is strictly increasing in (0; =2): Applying Lemma 1, we conclude that  is strictly increasing in (0; r): Hence,  (0+) <  (x) <  (r) and the desired p p p inequalities (3.3) can be obtained from this and the limits  (0+) = 1=(2p ) 1=6 and  (r) = ln [r=[(sinh r)(cos(r=p))]] =r : The proof is completed. 4. Some particular cases In this section, we obtain some sharp inequalities from our main results by assigning appropriate values to a parameter p therein. We list the inequalities for sinc and hyperbolic sinc function as follows. Putting p = 3 in Corollary 1 gives x sin x x (4.1) cos p < < cos p e ; 0 < x  ; x 2 3 3 where = (4= ) ln 2=( cos(=2 3))  0:013219: This includes the left inequality of (1.1). Putting p = 2 in Corollary 2, we obtain x  2 sin x x 2 x x =24 (4.2) cos e < < cos e ; 0 < x  ; 2 x 2 2 where = (4= ) ln 2 2=  0:042558: Lower and upper bounds of (4.2) are sharper than the corresponding lower and upper bounds of (1.1) in the intervals (&; =2] and (0; ), respectively, where   1:5204 and &  0:4633: An upper bound of (4.2) is sharper than that of (4.1) in (0;  ); where   1:5346: 1 1 The double inequality (4.2) is a complete refinement of (1.3) and (1.6) and it also refines corresponding lower and upper bound of (1.4) in the intervals (& ; =2) and (0;  ), respectively, where &  0:705 and   1:4372: Some of 1 2 1 2 12 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Visual comparison for lower bounds of (sin x)/x (sin x)/x and the lower bounds (sin x)/x cos(x/ 3 ) (cos(x 2)) 2 2 exp((4 log(2/π)/π )x ) 2 2 cos(x/2) exp(((4/π ) log(2 2 π))x ) 0.80 0.85 0.90 0.95 1.00 Figure 1. Visual comparison of lower bounds for (sin x)=x with x 2 [0:8; 1]; the obtained lower bound is in lightblue color these facts are illustrated in Figure 1 and Figure 2 for the lower and upper bounds of (sin x)=x, respectively. Visual comparison for upper bounds of (sin x)/x (sin x)/x and the upper bounds (sin x)/x cos((2/π) arcos(2/π) x) (cos(x 3)) exp(- x 6) cos(x/2) exp(- x 24) 0.80 0.81 0.82 0.83 0.84 0.85 Figure 2. Visual comparison of upper bounds for (sin x)=x with x 2 [0:8; 0:85]; the obtained upper bound is in light blue color 0.884 0.886 0.888 0.890 0.892 0.894 0.896 0.84 0.85 0.86 0.87 0.88 0.89 Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 13 From Figure 1 and Figure 2, it is clear that the obtained bounds for (sin x)=x significantly improve some established bounds of the literature. Putting p = 2 in Theorem 2 yields x 2 sinh x x 2 x x =24 (4.3) cosh e < < cosh e ; 0 < x  r; 2 x 2 where = ln [(sinh r)=(r cosh(r=2))] =r : The inequalities (4.3) uniformly refine (1.7). An upper bound of (4.3) is also a uniform refinement of (1.5). The lower bound of (4.3) is better than the corresponding lower bounds in (1.2) and (1.5) for smaller values of r: However, there is no strict comparison in this case for (0; r): Several other inequalities can be obtained and compared with existing inequalities. Figure 3 illustrates the sharpness of the obtained upper bound. From Fig- ure 3, we see that the gain of the obtained upper bound in the sharpness sense is consequent. Visual comparison for upper bounds of (sinh x)/x (sinh x)/x and the upper bounds (sinh x)/x (cosh x) exp(x 6) cosh(x 2) exp(x 24) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 3. Visual comparison of lower bounds for (sinh x)=x with x 2 [0; 3]; the obtained upper bound is in blue color Note. Due to the symmetry of the functions involved all the inequalities which are true in (0; ) are also true in (; 0): 1.0 1.5 2.0 2.5 3.0 14 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude References [1] G.D. Anderson, M.K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, In- equalities and Quasiconformal Maps, John Wiley & Sons, New York, 1997. [2] Y.J. Bagul and C. Chesneau, Refined forms of Oppenheim and Cusa-Huygens type inequalities, Acta Comment. Univ. Tartu. Math. 24 (2020), no. 2, 183–194. [3] Y.J. Bagul, R.M. Dhaigude, M. Kostić, and C. Chesneau, Polynomial-exponential bounds for some trigonometric and hyperbolic functions, Axioms 10 (2021), no. 4, Paper No. 308, 10 pp. [4] B. Chaouchi, V.E. Fedorov, and M. Kostić, Monotonicity of certain classes of functions related with Cusa-Huygens inequality, Chelyab. Fiz.-Mat. Zh. 6 (2021), no. 3, 31–337. [5] X.-D. Chen, H. Wang, J. Yu, Z. Cheng, and P. Zhu, New bounds of Sinc function by using a family of exponential functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 16, 17 pp. [6] C. Chesneau and Y.J. Bagul, A note on some new bounds for trigonometric functions using infinite products, Malays. J. Math. Sci. 14 (2020), no. 2, 273–283. [7] A.R. Chouikha, C. Chesneau, and Y.J. Bagul, Some refinements of well-known inequal- ities involving trigonometric functions, J. Ramanujan Math. Soc. 36 (2021), no. 3, 193–202. [8] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Seventh edition, Elsevier/Academic Press, Amsterdam, 2007. [9] K.S.K. Iyengar, B.S. Madhava Rao, and T.S. Nanjundiah, Some trigonometrical in- equalities, Half-Yearly J. Mysore Univ. Sect. B., N.S. 6 (1945), 1–12. [10] R. Klén, M. Visuri, and M. Vuorinen, On Jordan type inequalities for hyperbolic func- tions, J. Inequal. Appl. 2010, Art. ID 362548, 14 pp. [11] Y. Lv, G. Wang, and Y. Chu, A note on Jordan type inequalities for hyperbolic func- tions, Appl. Math. Lett. 25 (2012), no. 3, 505–508. [12] B. Malešević, T. Lutovac, and B. Banjac, One method for proving some classes of exponential analytical inequalities, Filomat 32 (2018), no. 20, 6921–6925. [13] B. Malešević and B. Mihailović, A minimax approximant in the theory of analytic inequalities, Appl. Anal. Discrete Math. 15 (2021), no. 2, 486–509. [14] D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970. [15] E. Neuman and J. Sándor, On some inequalities involving trigonometric and hyper- bolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl. 13 (2010), no. 4, 715–723. [16] E. Neuman and J. Sándor, Inequalities for hyperbolic functions, Appl. Math. Comput. 218 (2012), no. 18, 9291–9295. [17] C. Qian, X.-D. Chen, and B. Malesevic, Tighter bounds for the inequalities of Sinc function based on reparameterization, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 29, 38 pp. [18] J. Sándor, Two applications of the Hadamard integral inequality, Notes Number Theory Discrete Math. 23 (2017), no. 4, 52–55. [19] S. Wu and L. Debnath, Wilker-type inequalities for hyperbolic functions, Appl. Math. Lett. 25 (2012), no. 5, 837–842. [20] Z.-H. Yang, New sharp bounds for logarithmic mean and identric mean, J. Inequal. Appl. 2013, 2013:116, 17 pp. [21] Z.-H. Yang, Refinements of a two-sided inequality for trigonometric functions, J. Math. Inequal. 7 (2013), no. 4, 601–615. [22] Z.-H. Yang and Y.-M. Chu, Jordan type inequalities for hyperbolic functions and their applications, J. Funct. Spaces 2015, Art. ID 370979, 4 pp. [23] L. Zhang and X. Ma, Some new results of Mitrinović-Cusa’s and related inequalities based on the interpolation and approximation method, J. Math. 2021, Art. ID 5595650, 13 pp. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 15 [24] L. Zhu, Generalized Lazarevic’s inequality and its applications–Part II, J. Inequal. Appl. 2009, Art. ID 379142, 4 pp. [25] L. Zhu, Some new bounds for Sinc function by simultaneous approximation of the base and exponential functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 81, 17 pp. [26] L. Zhu and R. Zhang, New inequalities of Mitrinović-Adamović type, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 34, 15 pp. Yogesh J. Bagul Department of Mathematics K. K. M. College, Manwath Dist: Parbhani(M.S.)–431505 India e-mail: yjbagul@gmail.com Sumedh B. Thool Department of Mathematics Government Vidarbha Institute of Science and Humanities Amravati(M. S.)–444604 India e-mail: sumedhmaths@gmail.com Christophe Chesneau LMNO University of Caen-Normandie Caen France e-mail: christophe.chesneau@unicaen.fr Ramkrishna M. Dhaigude Department of Mathematics Government Vidarbha Institute of Science and Humanities Amravati(M. S.)–444604 India e-mail: rmdhaigude@gmail.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Mathematicae Silesianae de Gruyter

Refinements of Some Classical Inequalities Involving Sinc and Hyperbolic Sinc Functions

Download PDF
15 pages

Loading next page...
 
/lp/de-gruyter/refinements-of-some-classical-inequalities-involving-sinc-and-TPT0kW0vEM
Publisher
de Gruyter
Copyright
© 2023 Yogesh J. Bagul et al., published by Sciendo
eISSN
0860-2107
DOI
10.2478/amsil-2022-0019
Publisher site
See Article on Publisher Site

Abstract

Annales Mathematicae Silesianae 37 (2023), no. 1, 1–15 DOI: 10.2478/amsil-2022-0019 REFINEMENTS OF SOME CLASSICAL INEQUALITIES INVOLVING SINC AND HYPERBOLIC SINC FUNCTIONS Yogesh J. Bagul, Sumedh B. Thool , Christophe Chesneau, Ramkrishna M. Dhaigude Abstract. Several bounds of trigonometric-exponential and hyperbolic-expo- nential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions. 1. Introduction Consider the sinc function defined by sinc x = (sin x)=x; for x 6= 0 and sinc x = 1; for x = 0: A hyperbolic sinc function is defined similarly. Let us now cite some inequalities for sinc and hyperbolic sinc functions pertaining to the main results of this paper. First, the classical inequalities x sin x 2 2 (1.1) cos < < cos arccos  x ; 0 < x < ; x   2 were established by K.S.K. Iyengar, B.S. Madhava Rao and T.S. Nanjundiah in a little-known paper [9]. See also [14]. Recently, J. Sándor ([18]) offered Received: 25.03.2022. Accepted: 02.11.2022. Published online: 23.11.2022. (2020) Mathematics Subject Classification: 26D05, 26D07, 33B10. Key words and phrases: trigonometric-exponential, hyperbolic-exponential, Mitrinović- Adamović inequality, Lazarević inequality, Iyengar-Madhava Rao-Nanjudiah inequality. c 2022 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). 2 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude a new proof to the left inequality of (1.1) and proved its hyperbolic counterpart as follows: x sinh x (1.2) cosh p < ; x > 0: R. Klén, M. Visuri, and M. Vuorinen ([10]) found the following inequalities h  i h  i 2 3 x sin x x (1.3) cos < < cos ; 0 < x < : 2 x 3 2 Y. Lv, G. Wang, and Y. Chu ([11]) obtained the following: h  i h  i 4=3 a x sin x x (1.4) cos < < cos ; 0 < x < ; 2 x 2 2 where a = (ln(=2))=(ln 2)  1:30299: The left inequality of (1.4) is sharper than the corresponding left inequality of (1.3), whereas the right inequality of (1.3) is better than that of (1.4). The analogous inequality to (1.4) is the following one: h  i 4=3 x sinh x (1.5) cosh < < cosh x; x > 0; 2 x which can be seen in [15,16,20]. Exponential-type bounds for sinc and hyper- bolic sinc functions were obtained by Chesneau and Bagul in [6]. They are given below. We have 2 2 sin x x x =6 (1.6) e < < e ; 0 < x < ; x 2 where = 4 ln(2=)= , and 2 2 sinh x x x =6 (1.7) e < < e ; 0 < x < r; where r > 0 and  = ln[(sinh r)=r]=r : For different refinements, generalizations, and recent developments regard- ing inequalities involving the sinc and hyperbolic sinc functions, we refer the reader to [2–5,7,12,13,17,21–26]. This article aims to present new generalized bounds for sinc and hyperbolic sinc functions. Our bounds are trigonometric- exponential and hyperbolic-exponential in nature and they refine some exist- ing bounds in the literature. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 3 We consider the following plan: Section 2 presents some preliminaries and lemmas. The main results are given in Section 3. Section 4 ends the article with some particular cases and discussions. 2. Preliminaries and lemmas The following power series expansions involving Bernoulli numbers can be found in [8, 1.411]: 2n 2n 2 (2 1) 2n1 (2.1) tan x = jB jx ; jxj < ; 2n (2n)! 2 n=1 2n 1 2 2n1 (2.2) cot x = jB jx ; jxj < ; 2n x (2n)! n=1 and 2n 2n 2 (2 1) 2n1 (2.3) tanh x = B x ; jxj < ; 2n (2n)! 2 n=1 where B are the even indexed Bernoulli numbers. 2n We will also use the following l’Hôpital’s rule of monotonicity. Lemma 1 (l’Hôpital’s rule of monotonicity [1]). Let f; g : [a; b] ! R be two continuous functions which are differentiable on (a; b) and g 6= 0 on (a; b): 0 0 If f =g is increasing (or decreasing) on (a; b); then the functions (f(x) f(a))=(g(x) g(a)) and (f(x) f(b))=(g(x) g(b)) are also increasing (or 0 0 decreasing) on (a; b): If f =g is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 2 ([2, Lemma 4]). For x > 0; the function sinh x x cosh x k(x) = x sinh x is strictly increasing. Additionally, we prove the following auxiliary results which can be of in- dependent interest. 4 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Lemma 3 (Sharp upper bound for hyperbolic sinc). For x 6= 0; it is true that sinh x (x + sinh x)(1 + cosh x) (2.4) < : x 4x Proof. Due to the symmetry of the functions involved at both sides, it suffices to prove (2.4) for x > 0: We first consider f(x) = x + x sinh x 4 cosh x + 4: Then f (x) = 2x + x cosh x 3 sinh x > 0; due to well-known Cusa–Huygens inequality ([2]). Therefore, f(x) is increasing for x > 0 and we get f(x) > f(0), i.e., x + x sinh x 4(cosh x 1) > 0; which can be written as 4(cosh x 1)(cosh x + 1) < x(x + sinh x)(1 + cosh x) or 4 sinh x < x(x + sinh x)(1 + cosh x): This gives the required inequality (2.4). Remark 1. For x 6= 0; it is not difficult to prove (x + sinh x)(1 + cosh x) 2 + cosh x < : 4x 3 Thus, we have sinh x (x + sinh x)(1 + cosh x) 2 + cosh x (2.5) < < ; x 6= 0: x 4x 3 Remark 2. A double inequality analogous to (2.5) also holds in the case of trigonometric functions. It is stated as sin x (x + sin x)(1 + cos x) 2 + cos x (2.6) < < ; x 6= 0: x 4x 3 We skip the proof of (2.6) because it is very similar to that of (2.5). Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 5 Lemma 4 (Refined lower bound for Wilker-type inequality). For x 6= 0; it is true that x x x(cosh x 1)(sinh x x) (2.7) + > 2 + sinh x tanh x 2 sinh x x(sinh x x) = 2 + > 2: 2(1 + cosh x) Proof. It is enough to prove that x x x(cosh x 1)(sinh x x) + > 2 + : sinh x tanh x 2 sinh x Equivalently, it corresponds to 2x + 2x sinh x cosh x > 4 sinh x + x(cosh x 1)(sinh x x) or 2 2 x + x sinh x cosh x 4 sinh x + x sinh x + x cosh x > 0; i.e., x(x + sinh x)(1 + cosh x) > 4 sinh x; which is true by Lemma 3. Remark 3. The inequality (2.7) is a refinement of the Wilker-type in- equality for hyperbolic functions established by Wu and Debnath ([19]). Remark 4. It is interesting to see that the circular counterpart of (2.7) is also true for all non-zero real numbers. It is stated as follows: x x x(cos x 1)(sin x x) (2.8) + > 2 + sin x tan x 2 sin x x(x sin x) = 2 + > 2; x 6= 0: 2(1 + cos x) The proof of (2.8) is quite similar to that of (2.7). The importance of (2.7) lies in the fact that it is the sharpest Wilker-type inequality of its kind so far in the literature, and it holds for all non-zero real numbers, although its sharpness can be observed in (0; ) and (; 0) only. 6 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Lemma 5. For x 6= 0; it is true that x x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] + > 2 + sinh x tanh x p sinh (2x=p) if p  2: Proof. Let g(x) = 2 + x tanh x (x sech x) : Differentiation yields 0 2 2 2 g (x) = tanh x + 2x sech x tanh x x sech x x sinh x 1 2 2 = + 2x sech x tanh x > 0: cosh x x cosh x Hence, g(x) is increasing and we have that g(x=2)  g(x=p) if x=2  x=p, i.e., p  2: Now, we have x x x x x g = 2 + tanh sech 2 2 2 2 2 x x sinh(x=2) = 2 + 2 cosh(x=2) 4 cosh (x=2) x x sinh x = 2 + 2(1 + cosh x) 2(1 + cosh x) x(sinh x x) = 2 + 2(1 + cosh x) x(cosh x 1)(sinh x x) = 2 + : 2 sinh x Similarly, we establish that x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] g = 2 + : p sinh (2x=p) By making use of Lemma 4, the conclusion of Lemma 5 follows.  Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 7 3. Main results We are now in a position to state and prove our main results. Theorem 1. For p > 1; we define  : (0; =2] ! R by h i sin x ln x cos(x=p) (x) = : Then 1.  is strictly increasing if p  3; 2.  is strictly decreasing if p  2: Proof. We write ln [(sin x)=x] ln[cos(x=p)] ( ) (x) 1 p (x) = = ; x  (x) where ( ) (x) = ln [(sin x)=x]ln[cos(x=p)] and  (x) = x , with ( ) (0+) = 1 p 2 1 p 0 =  (0): By differentiation, we get ( ) (x) 1 x cos x sin x tan(x=p) = p + (x) 2p x sin x x 1 tan(x=p) cot x p = + p : 2p x x x Utilizing (2.1) and (2.2), we obtain " # 1 1 2n 2n 2n X X ( ) (x) 1 2 (2 1) p 2 2n2 2n2 = jB jx jB jx 2n 2n 0 2n1 (x) 2p p  (2n)! (2n)! n=1 n=1 2n 2n 1 2 2 1 2n2 = p jB jx : 2n 2n1 2p (2n)! p n=1 2n 2n1 By Lemma 1,  will be strictly increasing if (2 1)=p p > 0, i.e., 2n 2n 2n 1=2n 2 1 > p or p < h(n) := (2 1) : And it is easy to show that h(n) is strictly increasing for n = 1; 2; : This implies that p  inffh(n) : n = 1; 2;g = h(1) = 3: Similarly, we can say that  will be strictly 2n 2n1 decreasing if we have (2 1)=p p < 0; or p > h(n): So, we get p  supfh(n) : n = 1; 2;g = lim h(n) = 2: This completes the proof n!1 of Theorem 1.  8 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Next, by l’Hôpital’s rule,  (0+) = lim  (x) = 1=(2p ) 1=6; and p x!0 p (=2) = (4= ) ln [2=[ cos(=(2p))]] : Hence, we immediately deduce the following corollaries: Corollary 1. If 1 < p  3 and 0 < x  =2; then the best possible constants and such that the inequalities 1 1 x 2 sin x x 2 x x 1 1 cos e < < cos e p x p 2 2 hold are 1=(2p ) 1=6 and (4= ) ln [2=[ cos(=(2p))]], respectively. Corollary 2. If p  2 and 0 < x  =2; then the inequalities 2 2 x sin x x x x 1 1 cos e < < cos e p x p hold with the best possible constants and which are as defined in the 1 1 Corollary 1. An analogous result involving hyperbolic functions is formulated in the following theorem. Theorem 2. For p > 0 and r > 0; we define a function : (0; r) ! R by h i sinh x ln x cosh(x=p) (x) = : Then is strictly decreasing if p  2: In particular, if p  2; then the best possible constants and such that the inequalities 2 2 x 2 sinh x x 2 x x 2 2 (3.1) cosh e < < cosh e ; 0 < x < r; p x p 2 2 hold are ln ((sinh r)=[r cosh(r=p)]) =r and 1=6 1=(2p ), respectively. Proof. Set ( ) (x) = ln [(sinh x)=x] ln [cosh(x=p)] and (x) = x : 1 p 2 Clearly ( ) (0+) = 0 = (0) and (x) = ( ) (x)= (x): In view of using 1 p 2 p 1 p 2 Lemma 1, we differentiate and obtain ( ) (x) 1 coth x p tanh(x=p) 1 = p := ( ) (x): 3 p (x) 2p x x x 2p 2 Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 9 Then, we get p p 2p 1 x 1 x 0 2 2 ( ) (x) = cosech x coth x + sech + tanh 2 3 2 x x x px p x p p x x x x x x = + 2 + sech tanh x sinh x tanh x p p p p p x x = + 2 x sinh x tanh x x[cosh(2x=p) 1][sinh(2x=p) 2x=p] p sinh (2x=p) By Lemma 1 and Lemma 5, we conclude that is strictly decreasing if p  2: Consequently, (0+) > (x) > (r); 0 < x < r: p p p The desired inequalities (3.1) follow due to the limits (0+) = 1=6 1=(2p ) and (r) = ln [(sinh r)=[r cosh(r=p)]] =r : Theorem 3. For p > 1; we define ' : (0; =2] ! R by h i sin x ln x cosh(x=p) ' (x) = : Then ' is strictly decreasing if p  2: In particular, if p  2; then the best possible constants and such that the inequalities 3 3 x 2 sin x x 2 x x 3 3 (3.2) cosh e < < cosh e ; 0 < x p x p 2 2 2 hold are (4= ) ln [2=[ cosh(=(2p))]] and 1=(2p ) + 1=6 , respectively. Proof. We begin with ln (sin x=x) ln(cosh x=p) (' ) (x) 1 p ' (x) = = ; x ' (x) 2 10 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude where (' ) (x) = ln [(sin x)=x]ln[cosh(x=p)] and ' (x) = x with (' ) (0+) = 1 p 2 1 p 0 = ' (0): Differentiation gives (' ) (x) 1 x cos x sin x tanh(x=p) = p ' (x) 2p x sin x x 1 cot x p tanh(x=p) = p : 2p x x x Utilizing (2.1) and (2.3), we obtain " # 1 1 2n 2n 2n X X (' ) (x) 1 p 2 2 (2 1) 2n2 2n2 = jB jx B x 2n 2n 2n1 ' (x) 2p (2n)! p  (2n)! n=1 n=1 2n 2n 1 2 (2 1) 2n2 = pjB j B x 2n 2n 2n1 2p (2n)! p n=1 2n 1 2 2n2 := a x ; 2p (2n)! n=1 2n 2n1 where a = pjB j [(2 1)=p ]B : By Lemma 1, ' will be strictly n 2n 2n p decreasing if a > 0: But, a is always positive for B < 0 irrespective of n n 2n p: So we consider the case when B > 0: In this case, a > 0 implies that 2n n 2n 2n 2n 2n 2n 1=(2n) jB j > [(2 1)=p ]B or p > (2 1), i.e., p > (2 1) := h(n) 2n 2n and h(n) being strictly increasing, we write p  supfh(n) : n = 1; 2;g = lim h(n) = 2: Finally, ' (0+) > ' (x) > ' (=2), and the limits n!1 p p p 2 2 ' (0+) = 1=(2p ) + 1=6 and ' (=2) = (4= ) ln [2=[ cosh(=(2p))]] p p give the inequalities (3.2). Theorem 4. For p  1 and r < p=2; we define  : (0; r) ! R by h i ln sinh x cos(x=p) (x) = : Then  is strictly increasing. In particular, if p  1; then the best possible constants and such that the inequalities 4 4 2 2 x x 4 4 e sinh x e (3.3) < < ; 0 < x < r cos(x=p) x cos(x=p) 2 2 hold are ln [r=[(sinh r)(cos(r=p))]] =r and 1=6 1=(2p ), respectively. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 11 Proof. We have ( ) (x) ( ) (0+) 1 p 1 p (x) = ; (x)  (0) 2 2 where ( ) (x) = ln (x=sinh x)ln [cos(x=p)] and  (x) = x with ( ) (0+) = 1 p 2 1 p 0 =  (0): Differentiation yields ( ) (x) 1 sinh x x cosh x 1 tan(x=p) = + ; 2 2 (x) 2 x sinh x 2p (x=p) which is strictly increasing because of Lemma 2 and the fact that (tan x)=x is strictly increasing in (0; =2): Applying Lemma 1, we conclude that  is strictly increasing in (0; r): Hence,  (0+) <  (x) <  (r) and the desired p p p inequalities (3.3) can be obtained from this and the limits  (0+) = 1=(2p ) 1=6 and  (r) = ln [r=[(sinh r)(cos(r=p))]] =r : The proof is completed. 4. Some particular cases In this section, we obtain some sharp inequalities from our main results by assigning appropriate values to a parameter p therein. We list the inequalities for sinc and hyperbolic sinc function as follows. Putting p = 3 in Corollary 1 gives x sin x x (4.1) cos p < < cos p e ; 0 < x  ; x 2 3 3 where = (4= ) ln 2=( cos(=2 3))  0:013219: This includes the left inequality of (1.1). Putting p = 2 in Corollary 2, we obtain x  2 sin x x 2 x x =24 (4.2) cos e < < cos e ; 0 < x  ; 2 x 2 2 where = (4= ) ln 2 2=  0:042558: Lower and upper bounds of (4.2) are sharper than the corresponding lower and upper bounds of (1.1) in the intervals (&; =2] and (0; ), respectively, where   1:5204 and &  0:4633: An upper bound of (4.2) is sharper than that of (4.1) in (0;  ); where   1:5346: 1 1 The double inequality (4.2) is a complete refinement of (1.3) and (1.6) and it also refines corresponding lower and upper bound of (1.4) in the intervals (& ; =2) and (0;  ), respectively, where &  0:705 and   1:4372: Some of 1 2 1 2 12 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude Visual comparison for lower bounds of (sin x)/x (sin x)/x and the lower bounds (sin x)/x cos(x/ 3 ) (cos(x 2)) 2 2 exp((4 log(2/π)/π )x ) 2 2 cos(x/2) exp(((4/π ) log(2 2 π))x ) 0.80 0.85 0.90 0.95 1.00 Figure 1. Visual comparison of lower bounds for (sin x)=x with x 2 [0:8; 1]; the obtained lower bound is in lightblue color these facts are illustrated in Figure 1 and Figure 2 for the lower and upper bounds of (sin x)=x, respectively. Visual comparison for upper bounds of (sin x)/x (sin x)/x and the upper bounds (sin x)/x cos((2/π) arcos(2/π) x) (cos(x 3)) exp(- x 6) cos(x/2) exp(- x 24) 0.80 0.81 0.82 0.83 0.84 0.85 Figure 2. Visual comparison of upper bounds for (sin x)=x with x 2 [0:8; 0:85]; the obtained upper bound is in light blue color 0.884 0.886 0.888 0.890 0.892 0.894 0.896 0.84 0.85 0.86 0.87 0.88 0.89 Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 13 From Figure 1 and Figure 2, it is clear that the obtained bounds for (sin x)=x significantly improve some established bounds of the literature. Putting p = 2 in Theorem 2 yields x 2 sinh x x 2 x x =24 (4.3) cosh e < < cosh e ; 0 < x  r; 2 x 2 where = ln [(sinh r)=(r cosh(r=2))] =r : The inequalities (4.3) uniformly refine (1.7). An upper bound of (4.3) is also a uniform refinement of (1.5). The lower bound of (4.3) is better than the corresponding lower bounds in (1.2) and (1.5) for smaller values of r: However, there is no strict comparison in this case for (0; r): Several other inequalities can be obtained and compared with existing inequalities. Figure 3 illustrates the sharpness of the obtained upper bound. From Fig- ure 3, we see that the gain of the obtained upper bound in the sharpness sense is consequent. Visual comparison for upper bounds of (sinh x)/x (sinh x)/x and the upper bounds (sinh x)/x (cosh x) exp(x 6) cosh(x 2) exp(x 24) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 3. Visual comparison of lower bounds for (sinh x)=x with x 2 [0; 3]; the obtained upper bound is in blue color Note. Due to the symmetry of the functions involved all the inequalities which are true in (0; ) are also true in (; 0): 1.0 1.5 2.0 2.5 3.0 14 Yogesh J. Bagul, Sumedh B. Thool, Christophe Chesneau, Ramkrishna M. Dhaigude References [1] G.D. Anderson, M.K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, In- equalities and Quasiconformal Maps, John Wiley & Sons, New York, 1997. [2] Y.J. Bagul and C. Chesneau, Refined forms of Oppenheim and Cusa-Huygens type inequalities, Acta Comment. Univ. Tartu. Math. 24 (2020), no. 2, 183–194. [3] Y.J. Bagul, R.M. Dhaigude, M. Kostić, and C. Chesneau, Polynomial-exponential bounds for some trigonometric and hyperbolic functions, Axioms 10 (2021), no. 4, Paper No. 308, 10 pp. [4] B. Chaouchi, V.E. Fedorov, and M. Kostić, Monotonicity of certain classes of functions related with Cusa-Huygens inequality, Chelyab. Fiz.-Mat. Zh. 6 (2021), no. 3, 31–337. [5] X.-D. Chen, H. Wang, J. Yu, Z. Cheng, and P. Zhu, New bounds of Sinc function by using a family of exponential functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 16, 17 pp. [6] C. Chesneau and Y.J. Bagul, A note on some new bounds for trigonometric functions using infinite products, Malays. J. Math. Sci. 14 (2020), no. 2, 273–283. [7] A.R. Chouikha, C. Chesneau, and Y.J. Bagul, Some refinements of well-known inequal- ities involving trigonometric functions, J. Ramanujan Math. Soc. 36 (2021), no. 3, 193–202. [8] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Seventh edition, Elsevier/Academic Press, Amsterdam, 2007. [9] K.S.K. Iyengar, B.S. Madhava Rao, and T.S. Nanjundiah, Some trigonometrical in- equalities, Half-Yearly J. Mysore Univ. Sect. B., N.S. 6 (1945), 1–12. [10] R. Klén, M. Visuri, and M. Vuorinen, On Jordan type inequalities for hyperbolic func- tions, J. Inequal. Appl. 2010, Art. ID 362548, 14 pp. [11] Y. Lv, G. Wang, and Y. Chu, A note on Jordan type inequalities for hyperbolic func- tions, Appl. Math. Lett. 25 (2012), no. 3, 505–508. [12] B. Malešević, T. Lutovac, and B. Banjac, One method for proving some classes of exponential analytical inequalities, Filomat 32 (2018), no. 20, 6921–6925. [13] B. Malešević and B. Mihailović, A minimax approximant in the theory of analytic inequalities, Appl. Anal. Discrete Math. 15 (2021), no. 2, 486–509. [14] D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970. [15] E. Neuman and J. Sándor, On some inequalities involving trigonometric and hyper- bolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl. 13 (2010), no. 4, 715–723. [16] E. Neuman and J. Sándor, Inequalities for hyperbolic functions, Appl. Math. Comput. 218 (2012), no. 18, 9291–9295. [17] C. Qian, X.-D. Chen, and B. Malesevic, Tighter bounds for the inequalities of Sinc function based on reparameterization, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 29, 38 pp. [18] J. Sándor, Two applications of the Hadamard integral inequality, Notes Number Theory Discrete Math. 23 (2017), no. 4, 52–55. [19] S. Wu and L. Debnath, Wilker-type inequalities for hyperbolic functions, Appl. Math. Lett. 25 (2012), no. 5, 837–842. [20] Z.-H. Yang, New sharp bounds for logarithmic mean and identric mean, J. Inequal. Appl. 2013, 2013:116, 17 pp. [21] Z.-H. Yang, Refinements of a two-sided inequality for trigonometric functions, J. Math. Inequal. 7 (2013), no. 4, 601–615. [22] Z.-H. Yang and Y.-M. Chu, Jordan type inequalities for hyperbolic functions and their applications, J. Funct. Spaces 2015, Art. ID 370979, 4 pp. [23] L. Zhang and X. Ma, Some new results of Mitrinović-Cusa’s and related inequalities based on the interpolation and approximation method, J. Math. 2021, Art. ID 5595650, 13 pp. Refinements of some classical inequalities involving sinc and hyperbolic sinc functions 15 [24] L. Zhu, Generalized Lazarevic’s inequality and its applications–Part II, J. Inequal. Appl. 2009, Art. ID 379142, 4 pp. [25] L. Zhu, Some new bounds for Sinc function by simultaneous approximation of the base and exponential functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 81, 17 pp. [26] L. Zhu and R. Zhang, New inequalities of Mitrinović-Adamović type, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 34, 15 pp. Yogesh J. Bagul Department of Mathematics K. K. M. College, Manwath Dist: Parbhani(M.S.)–431505 India e-mail: yjbagul@gmail.com Sumedh B. Thool Department of Mathematics Government Vidarbha Institute of Science and Humanities Amravati(M. S.)–444604 India e-mail: sumedhmaths@gmail.com Christophe Chesneau LMNO University of Caen-Normandie Caen France e-mail: christophe.chesneau@unicaen.fr Ramkrishna M. Dhaigude Department of Mathematics Government Vidarbha Institute of Science and Humanities Amravati(M. S.)–444604 India e-mail: rmdhaigude@gmail.com

Journal

Annales Mathematicae Silesianae – de Gruyter

Published: Mar 1, 2023

Keywords: trigonometric-exponential; hyperbolic-exponential; Mitrinović-Adamović inequality; Lazarević inequality; Iyengar-Madhava Rao-Nanjudiah inequality; 26D05; 26D07; 33B10

There are no references for this article.