Abstract
We present the best possible parametersp,q∈0,∞such that the double inequality1/3p2cosh(px)+1-1/3p2<sinh(x)/x<1/3q2cosh(qx) + 1 - 1/3q2holds for allx∈0,∞. As applications, some new inequalities for certain special function and bivariate means are found.
Highlights
The well known Jordan inequality [1] is given by 2 π x x, x ∈ (0, π ) . 2 (1)
The hyperbolic counterpart and its generalizations have been the subject of intensive research
Zhu [14] proved that the inequality holds for all x > 0 if and only if q ≥ 3(1 − p) if p ∈ (−∞, 8/15] ∪ (1, ∞)
Summary
During the past few years, the improvements, refinements, and generalizations for inequality (1) have attracted the attention of many researchers [2–13]. The hyperbolic counterpart and its generalizations have been the subject of intensive research. Holds for all x ∈ (0, 1) if and only if p ≤ 1/3 and q ≥ [log(sinh(1))]/[log(cosh(1))] = 0.3721 ⋅ ⋅ ⋅. Holds for all x > 0 if and only if p ≥ √5/5 and q ≤ 1/3. The main purpose of this paper is to find the best possible parameters p, q ∈ (0, ∞) such that the double inequality (1/3p2) cosh(px) + 1 − 1/3p2 < sinh(x)/x < (1/3q2) cosh(qx) + 1 − 1/3q2 holds for all x > 0 and present several new inequalities for certain special function and bivariate means
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