MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

Guo, Bai-Ni; Guo, Bai-Ni; Liu, Ai-Qi; Liu, Ai-Qi; Qi, Feng; Qi, Feng

P-ISSN1226-0657
E-ISSN2287-6081
KCI

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Vol.15 No.4

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MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081

2008, v.15 no.4, pp.387-392

Guo, Bai-Ni
Liu, Ai-Qi
Qi, Feng

Guo,, B. , Liu,, A. , & Qi,, F. (2008). MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 15(4), 387-392.

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Abstract

In this note, an alternative proof and extensions are provided for the following conclusions in [6, Theorem 1 and Theorem 3]: The functions <TEX>$\frac1{x^2}-\frac{e^{-x}}{(1-e^{-x})^2}\;and\;\frac1{t}-\frac1{e^t-1}$</TEX> are decreasing in (0, <TEX>${\infty}$</TEX>) and the function <TEX>$\frac{t}{e^{at}-e^{(a-1)t}}$</TEX> for a <TEX>$a{\in}\mathbb{R}\;and\;t\;{\in}\;(0,\;{\infty})$</TEX> is logarithmically concave.

keywords: alternative proof, monotonicity, logarithmic convexity, exponential function, extension, Hermite-Hadamard's integral inequality, inequality, power series expansion

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Article Contents

Vol.15 No.4

MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

Abstract

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics