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ISSN : 1226-0657

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Vol.15 No.4
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MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

한국수학교육학회지시리즈B:순수및응용수학 / 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 (SCHOOL OF MATHEMATICS AND INFORMATICS, HENAN POLYTECHNIC UNIVERSITY)
Liu, Ai-Qi (DEPARTMENT OF MATHEMATICS, SANMENXIA POLYTECHNIC)
Qi, Feng (RESEARCH INSTITUTE OF MATHEMATICAL INEQUALITY THEORY, HENAN POLYTECHNIC UNIVERSITY)
Guo, Bai-Ni, Liu, Ai-Qi, & Qi, Feng. (2008). MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION. 한국수학교육학회지시리즈B:순수및응용수학, 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

한국수학교육학회지시리즈B:순수및응용수학

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