Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities

Journal of Inequalities and Applications, Dec 2016

In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality 2 β α + 2 β ( sin x x ) α + α α + 2 β ( tan x x ) β − 1 > ( < ) 0 holds for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

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Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities

Sun et al. Journal of Inequalities and Applications Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities Hui Sun 0 1 2 Zhen-Hang Yang 0 1 Yu-Ming Chu 0 1 2 0 University , Yiyang, 413000 , China 1 Computation Sciences , Hunan City 2 School of Mathematics In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality inequalities; 2β; Wilker-type inequality; sine function; tangent function; necessary and - – 1 > (<)0 MSC: 26D05; 33B10 sufficient condition 1 Introduction The Wilker inequality [, ] for sine and tangent functions states that the inequality (.) have been the subject of intensive research in the recent years. Wu and Srivastava [] proved that the inequality and Sándor [] generalized inequality (.) to the Bessel functions. In [], Zhu proved that the inequalities © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Zhu [] proved that the inequalities are valid for all x ∈ (, π /) if (p, λ, η) ∈ {(p, λ, η)|p ≥ , λ ≥  – (/π )p, η ≤ /} ∪ {(p, λ, η)| ≤ p ≤ /, λ ≥ /, η ≤  – (/π )p}. In [], Yang and Chu provided the necessary and sufficient condition for the parameter μ such that the generalized Wilker-type inequality –  > (<) holds for any fixed λ ≥  and all x ∈ (, π /). Very recently, Chu et al. [] proved that the two parameter generalized Wilker-type inequality ∪ (α, β) β > , –  ≤ α + β <   –  < α < , β ≤ –  , α + β +  ≤  , The main purpose of this paper is to provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality (.) and its reversed inequality hold for all x ∈ (, π /). 2 Lemmas holds for all x ∈ (α, β) if there exists η ∈ (α, β) such that f (x)/f(x) is strictly increasing (decreasing) on (α, η) and strictly decreasing (increasing) on (η, β), and F(x) = cos x(sin x – x cos x)(x – sin x cos x), G(x) = (x – sin x cos x)(sin x – x cos x), I(x) = log x – log(sin x), J(x) = log(tan x) – log x, Lemma . (See [], Lemma .) Let β ∈ R, x ∈ (, π /), and F(x), G(x), H(x) and g(x) be defined by respectively. Then it is not difficult to verify that Lemma . (See [], Lemma .) Let x ∈ (, π /) and Qα,β (x) be defined by (.). Then the following statements are true: () If α + β + / ≥  and β ≥ –, then Qα,β (x) is strictly decreasing on (, π /). () If α ≤ π / –  and –/ < β ≤ –, then Qα,β (x) is strictly increasing on (, π /). () If α + β + / ≤  and β ≤ –/, then Qα,β (x) is strictly increasing on (, π /). Lemma . Let x ∈ (, π /), Qα,β (x) be defined by (.) and the function x → D(α, β; x) be defined by D(α, β; x) = Qα,β (x) –  . Then the following statements are true: () If α ∈ R is fixed and β < , then there exists a unique solution β = β(α) given by = –. · · · . Proof Part () follows easily from (.)-(.) and the fact that [(/π )α – ]/α < . () It follows from (.) and (.) that = ∞, = –  . = –. · · · . () The function α → β(α) is strictly decreasing follows easily from (.) and (.). The function β → α(β) is strictly decreasing due to it is the inverse function of α → β(α). is the unique solution of the equation β(α) = –α/ – / such that β(α) < –α/ – / for α < α and β(α) > –α/ – / for α > α. Therefore, Qα,β (x) > / for all x ∈ (, π /) follows from Lemma .() and (.) together with (.). for α = , where the last of (.) due to log x – x +  <  for all x >  with x = . Inequality (.) implies that the function α → P(α) is strictly increasing on (, ∞). Therefore, there exists a unique α = α that satisfies the equation β(α) = –α/ – / such that β(α) < –α/ – / for α < α and β(α) > –α/ – / for α > α follows from (.) and the monotonicity of the function α → P(α). Numerical computations show that α = –. · · · . ≥ , which implies that β ≤ –α/ – /. If α ≥ –/ and β ≤ –α/ – /, then we clearly see for x ∈ (, π /) and β =  that the function β → Qα,β (x) is strictly decreasing. Therefore, it suffices to prove that Qα,β (x) < / for all x ∈ (, π /) if α ≥ α∗ and β = β(α). From (.) and Lemma .() we get which implies that β ≥ –α/ – /. If α ≤ –/ and β ≥ –α/ – /, then we clearly see that β = β(α) ≤ β α∗ = –  . cosβ x sinα–β– x = – (x – sin x cos x) g(x) + α F(x)xβ–α–, xl→im+ g(x) + α = α + β +  > , From (.) and (.) together with the monotonicity of the function x → g(x) + α on the interval (, π /) we clearly see that there exists x ∈ (, π /) such that the function x → Iα(x)/Jβ (x) is strictly decreasing on (, x) and strictly increasing on (x, π /). Note that Iα( π –) – Iα(+) π – Jβ(α)( π –) – Jβ(α)(+) = Qα,β(α)  +  =  . Therefore, Qα,β (x) < / for all x ∈ (, π /) follows from Lemma ., (.), (.), (.), and the piecewise monotonicity of the function x → Iα(x)/Jβ (x) on the interval (, π /). () If α ≤ –/ and Qα,β (x) < / for all x ∈ (, π /), then from (.)-(.) we have ≤ , Therefore, Qα,β (x) < / for x ∈ (, π /) follows easily from Lemma .(), (.), and (.). () If α ≤ α and Qα,β (x) > / for all x ∈ (, π /), then (.) and Lemma .() lead to the conclusion that D(α, β; π –) ≥  and β ≤ β(α). Next, we prove that Qα,β (x) > / for all x ∈ (, π /) if α ≤ α and β ≤ β(α). Since the function β → Qα,β (x) is strictly decreasing which was proved in part (), we only need to prove that Qα,β (x) > / for all x ∈ (, π /) if α ≤ α and β = β(α). It follows from Lemma .() and (), Lemma .(), Lemma ., and α ≤ α < α that β ≥ β(α) = – and the function g(x) + α is strictly increasing on (, π /) such that xl→im+ g(x) + α = α + β +  < , From (.), (.), and (.) we clearly see that there exists x∗ ∈ (, π /) such that the function x → Iα(x)/Jβ (x) is strictly increasing on (, x∗) and strictly decreasing on (x∗, π /). Therefore, Qα,β (x) > / for all x ∈ (, π /) follows from Lemma ., (.), (.), (.), and the piecewise monotonicity of the function x → Iα(x)/Jβ (x) on the interval (, π /). ≤ , Therefore, Qα,β (x) > / for all x ∈ (, π /) follows from Lemma .() and (.). () If β ≤ –/ and Qα,β (x) > / for all x ∈ (, π /), then from (.)-(.) we have ≥ , Therefore, the desired result follows from Lemma .() and (.). Next, we prove that Qα,β (x) > / for all x ∈ (, π /) if – ≤ β <  and α ≤ α(β). It follows from – ≤ β <  and α ≤ α(β) together with Lemma .() that 3 Main results Let α, β ∈ R with αβ(α + β) =  and Qα,β (x) be defined by (.), then we clearly see that the generalized Wilker-type inequality holds for all x ∈ (, π /) if and only if Qα,β (x) < / and αβ(α + β) >  or Qα,β (x) > / and αβ(α + β) < , while the generalized Wilker-type inequality Then the following statements are true: Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. 1. Wilker , JB: Problem E3306 . Am. Math. Mon. 96 ( 1 ), 55 ( 1989 ) 2. Sumner , JS, Jagers, AA, Vowe , M, Anglesio, J: Inequalities involving trigonometric functions . Am. Math. Mon . 98 ( 3 ), 264 - 267 ( 1991 ) 3. Wu , S-H , Srivastava , H-M : A weighted and exponential generalization of Wilker's inequality and its applications . Integral Transforms Spec. 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Hui Sun, Zhen-Hang Yang, Yu-Ming Chu. Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities, Journal of Inequalities and Applications, 2016, 322, DOI: 10.1186/s13660-016-1270-2