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Annales Mathematicae Silesianae
, Volume 37 (1): 15 – Mar 1, 2023

/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

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. 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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

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

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