Logarithmic mean inequality for generalized trigonometric and hyperbolic functions

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Logarithmic mean inequality for generalized trigonometric and hyperbolic functions Bhayo, Barkat; Yin, Li Bhayo, B., & Yin, L. (2015). Logarithmic mean inequality for generalized trigonometric and hyperbolic functions. Acta Universitatis Sapientiae: Mathematica, 6(2), 135-145. https://doi.org/10.1515/ausm-2015-0002 2015 Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 135–145 Logarithmic mean inequality for generalized trigonometric and hyperbolic functions Barkat Ali Bhayo Department of Mathematical Information Technology, University of Jyv¨askyl¨a, 40014 Jyv¨askyl¨a, Finland email: [email protected] Li Yin Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, 256603, China email: yinli [email protected] Abstract. In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean. 1 Introduction Recently, the study of the generalized trigonometric and generalized hyperbolic functions has got huge attention of numerous authors, and has appeared the huge number of papers involving the equalities and inequalities and basis properties of these function, e.g. see [7,8,9,6,10,13,14,18,23] and the references therein. These generalized trigonometric and generalized hyperbolic functions p-functions depending on the parameter p>1were introduced by Lindqvist [19] in 1995. These functions coincides with the usual functions for p=2. Thereafter Takesheu took one further step and generalized these function for two parameters p, q > 1, so-called (p, q)-functions. In [8], some convexity and concavity properties of p-functions were studied. Thereafter those results were extendedin[5] for two parameters in the sense of Power mean inequality. In this paper we study the convexity and concavity property of p-function with 2010 Mathematics Subject Classification: 33B10; 26D15; 26D99 Key words and phrases: logarithmic mean, generalized trigonometric and hyperbolic functions, inequalities, generalized convexity 135 DOI: 10.1515/ausm-2015-0002 136 Generalized trigonometric and hyperbolic functions respect Logarithmic mean. Before we formulate our main result we will define generalized trigonometric and hyperbolic functions customarily. The eigenfunction sinpof the so-called one-dimensional p-Laplacian problem [12] −Δpu=−|u|p−2u =λ|u|p−2u, u(0)=u(1)=0, p > 1, is the inverse function of F:(0, 1)→0, πp 2, defined as F(x)=arcsinp(x)=x 0 (1−tp)−1 pdt, where πp=2arcsinp(1)= 2 p1 0 (1−s)−1/ps1/p−1ds =2 pB1−1 p,1 p=2π psin π p, here B(., .)denotes the classical beta function. The function arcsinpis called the generalized inverse sine function, and coincides with usual inverse sine function for p=2. Similarly, the other generalized inverse trigonometric and hyperbolic functions arccosp:(0, 1)→ (0, πp/2),arctanp:(0, 1)→(0, bp),arcsinhp:(0, 1)→(0, cp),arctanhp:(0, 1)→ (0, ∞),where bp=1 2p ψ1+p 2p −ψ1 2p=2−1 pF1 p,1 p;1+1 p;1 2, cp=1 21 p F1, 1 p;1+1 p,1 2, aredefinedasfollows arccosp(x)=(1−xp)1 p 0 (1−tp)−1 pdt, arctanp(x)=x 0 (1+tp)−1dt, arcsinhp(x)=x 0 (1+tp)−1 pdt, arctanhp(x)=x 0 (1−tp)−1dt, where F(a, b;c;z)is Gaussian hypergeometric function [1]. The generalized cosine function is defined by d dx sinp(x)=cosp(x),x∈[0, πp/2]. B. A. Bhayo, L. Yin 137 It follows from the definition that cosp(x)=(1−(sinp(x))p)1/p , and |cosp(x)|p+|sinp(x)|p=1, x ∈R.(1) Clearly we get d dx cosp(x)=−cosp(x)2−psinp(x)p−1. The generalized tangent function tanpis defined by tanp(x)= sinp(x) cosp(x), and applying (1)weget d dx tanp(x)=1+tanp(x)p. For x∈(0, ∞), the inverse of generalized hyperbolic sine function sinhp(x) is defined by arcsinhp(x)=x 0 (1+tp)−1/pdt, and generalized hyperbolic cosine and tangent functions are defined by coshp(x)= d dx sinhp(x),tanhp(x)= sinhp(x) coshp(x), respectively. It follows from the definitions that |coshp(x)|p−|sinhp(x)|p=1. (2) From above definition and (2) we get the following derivative formulas, d dx coshp(x)=coshp(x)2−psinhp(x)p−1,d dx tanhp(x)=1−|tanhp(x)|p. Note that these generalized trigonometric and hyperbolic functions coincide with usual functions for p=2. For two distinct positive real numbers xand y, the Arithmetic mean, Geometric mean, Logarithmic mean, Harmonic mean and the Power mean of order p∈Rare respectively defined by A(x, y)=x+y 2,G(x, y)=√xy, 138 Generalized trigonometric and hyperbolic functions L(x, y)= x−y log(x)−log(y),x=y, H(x, y)= 1 A(1/x, 1/y), and Mt=⎧ ⎨ ⎩xt+yt 21/t ,t=0, √xy, t=0. Let f:I→(0, ∞)be continuous, where Iis a sub-interval of (0, ∞).Let Mand Nbe the means defined above, the we call that the function fis MNconvex (concave) if f(M(x, y)) ≤(≥)N(f(x),f(y)) for all x, y ∈I. Recently, Generalized convexity/concavity with respect to general mean values has been studied by Anderson et al. in [2]. We recall one of their results as follows Lemma 1 [2, Theorem 2.4] Let Ibe an open sub-interval of (0, ∞)and let f:I→(0, ∞)be differentiable. Then fis HH-convex (concave) on Iif and only if x2f(x)/f(x)2is increasing (decreasing). In [4], Baricz studied that if the functions fis differentiable, then it is (a, b)-convex (concave) on Iif and only if x1−af(x)/f(x)1−bis increasing (decreasing). It is important to mention that (1, 1)-convexity means the AA-convexity, (1, 0)-convexity means the AG-convexity, and (0, 0)-convexity means GG-convexity. Motivated by the results given in [2,4], we contribute to the topic by giving the following result. Theorem 1 Let f:I→(0, ∞)be a continuous and I⊆(0, ∞),then 1. L(f(x),f(y)) ≥(≤)f(L(x, y)), 2. L(f(x),f(y)) ≥(≤)f(A(x, y)), if fis increasing and log-convex (concave). Theorem 2 For x, y ∈(0, πp/2), the following inequalities 1. L(sinp(x),sinp(y)) ≤sinp(L(x, y)),p>1, B. A. Bhayo, L. Yin 139 2. L(cosp(x),cosp(y)) ≤cosp(L(x, y)),p≥2. Theorem 3 For p>1, we have 1. L(1/ sinp(x),1/sinp(y)) ≥1/ sinp(A(x, y)),x,y∈(0, πp/2), 2. L(1/ cosp(x),1/cosp(y)) ≥1/ cosp(L(x, y)),x,y∈(0, πp/2), 3. L(tanhp(x),tanhp(y)) ≤tanhp(A(x, y)),x,y∈(0, ∞), 4. L(arcsinhp(x),arcsinhp(y)) ≤arcsinhp(A(x, y)),x,y∈(0, 1), 5. L(arctanp(x),arctanp(y)) ≤arctanp(A(x, y)),x,y∈(0, 1). 2 Preliminaries and Proofs We give the following lemmas which will be used in the proof of our main result. Lemma 2 [22]Let f, g :[a, b]→Rbe integrable functions, both increasing or both decreasing. Furthermore, let p:[a, b]→Rbe a positive, integrable function. Then b a p(x)f(x)dx b a p(x)g(x)dx ≤b a p(x)dx b a p(x)f(x)g(x)dx. (3) If one of the functions for gis non-increasing and the other non-decreasing, then the inequality in (3) is reversed. Lemma 3 [17]If f(x)is continuous and convex function on [a, b],and ϕ(x) is continuous on [a, b],then f1 b−ab a ϕ(x)dx≤1 b−ab a f(ϕ(x))dx. (4) If function f(x)is continuous and concave on [a, b],then the inequality in (4) reverses. Lemma 4 [3]For two distinct positive real numbers a, b, we have L<A. Lemma 5 For p>1, the function sinp(x)is HH-concave on (0, πp/2). 140 Generalized trigonometric and hyperbolic functions Proof. Let f(x)=f1(x)f2(x),x ∈(0, πp/2),wheref1(x)=1/ sin(x)and f2(x)=x2cosp(x)/sinp(x). Clearly, f1is decreasing, so it is enough to prove that f2is decreasing, then the proof follows from Lemma 1.Weget f 2(x)=sinp(x)(cosp(x)−xcosp(x)2−psinp(x)p−1)−xcosp(x)2 sinp(x)2 =cosp(x)2((1−xtanp(x)p−1)tanp(x)−x) sinp(x)2=f3(x)cosp(x)2 sinp(x)2, where f3(x)=tanp(x)−xtanp(x)p−1. Again, one has f 3(x)=ptanp(x)p−1(1+tanp(x)p)x<0. Thus, f3is decreasing and g(x)<g(0)=0. This implies that f 2<0, hence f2is strictly decreasing, the product of two decreasing functions is decreasing. This implies the proof.  Proof of Theorem 1.We get L(f(x),f(y)) = f(x) f(y)1dt f(x) f(y) 1 tdt =x yf(u)du x y f(u) f(u)du .(5) It is assumed that the function f(x)is increasing and log fis convex, this implies that f(x) f(x)is increasing. Letting p(x)=1, f(x)=f(u)and g(x)= f(u)/f(u)in Lemma 2,weget x y 1du x y f(u)du ≥x y f(u) f(u)du x y f(u)du. This is equivalent to L(f(x),f(y)) = x yf(u)du x y f(u) f(u)du ≥x yf(u)du x y1du . By Lemmas 3and 4, and keeping in mind that log-convexity of fimplies the convexity of f,weget L(f(x),f(y)) ≥fx yudu x−y=fx+y 2≥f(L(x, y)). The proof of converse follows similarly. If we repeat the lines of proof of part (1), and use the concavity of the function, and Lemmas 3&4then we arrive at the proof of part (2). B. A. Bhayo, L. Yin 141 Proof of Theorem 2.It is easy to see that the function sinp(x)is increasing and log-concave. So the proof of part (1) follows easily from Theorem 1.We also offer another proof as follows: It can be observed easily that L(sinp(x),sinp(y))=x ycosp(u)du sinp(x) sinp(y) 1 tdt =x ycospudu x y cospu sinp(u)du, and sinp(L(x, y)) =sinpx−y log x y=sinpx y1du x y 1 udu. Clearly, cosp(u)and sinp(1/u), utilizing Chebyshev inequality, we have x y cosp(u)du x y sinp(1/u)du ≤x y 1du x y cospusinp 1 udu. So, we get x y cospudu x y sinp(1/u)du < x y 1du x y cosp(u) sinp(u)du. Where we apply simple inequality sinp1 u<1 sinp(u). In order to prove inequality (1), we only prove x y1du x ysinp(1/u)du ≤sinpx y1du x ysinp(1/u)du. Consider a partition Tof the interval [y, x]into nequal length sub-interval by means of points y=x0<x 1<···<x n=xand Δxi=x−y n. Picking an arbitrary point ξi∈[xi−1,x i]and using Lemma 1.2, we have n n  i=1 sinp1 ξi ≤sinp⎛ ⎜ ⎜ ⎝ n n  i=1 1 ξi ⎞ ⎟ ⎟ ⎠ ⇔x−y lim n→∞ x−y n n  i=1 sinp1 ξi≤sinp⎛ ⎜ ⎜ ⎝ x−y lim n→∞ x−y n n  i=1 1 ξi⎞ ⎟ ⎟ ⎠ 142 Generalized trigonometric and hyperbolic functions ⇔x y1du x ysinp(1/u)du ≤sinpx y1du x ysinp(1/u)du. This completes the proof. For (2), clearly cosp(x)is decreasing and tanp(x)p−1is increasing. One has (cosp(x)) =cosp(x)tanp(x)p−2(1−p+(2−p)tanp(x)p)<0, this implies that cosp(x)is concave on (0, πp/2). Using Tchebyshef inequality, we have x y 1du x y cosp(u)tanp(u)p−1du ≤x y cosp(u)du x y tanp(u)p−1du, which is equivalent to x ycosp(u)tanp(u)p−1du x ytanp(u)p−1du ≤x ycosp(u)du x y1du .(6) Substituting t=cosp(u)in (6), we get L(cosp(x),cosp(y)) = cosp(x) cosp(y)1dt cosp(x) cosp(y) 1 tdt =x ycosp(u)tanp(u)p−1du x ytanp(u)p−1du ≤x ycosp(u)du x y1du . Using Lemma 3and concavity of cosp(x), we obtain L(cosp(x),cospy)≤cospx yudu x−y=cospx+y 2≤cosp(L(x, y)). Proof of Theorem 3.Let g1(x)=1/ cosp(x),x∈(0, πp/2)and g2(x)= tanhp(x),x>0.Weget (log(g1(x))) =(p−1)tanp(x)p−2(1+tanp(x)p)>0, and (log(g2(x))) =1−tanhp(x)p tanhp(x)2((1−p)tanhp(x)p−1)<0. This implies that g1and g2are log-convex, clearly both functions are increasing, and log-convexity implies the convexity, so g1and g2are convex functions. Now the proof follows easily from Theorem 1. The rest of proof follows similarly.