# American Institute of Mathematical Sciences

July  2009, 8(4): 1231-1249. doi: 10.3934/cpaa.2009.8.1231

## Properties and applications of a function involving exponential functions

 1 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China 2 Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

Received  January 2008 Revised  January 2009 Published  March 2009

In the present paper, we give necessary and sufficient conditions for the elementary function $q_{\alpha,\beta}(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}},$ if $t\ne 0$ or $q_{\alpha,\beta}(t)=\beta-\alpha,$ if $t=0$ to be monotonic or logarithmically convex on $(-\infty,\infty)$, $(-\infty,0)$ or $(0,\infty)$ respectively, where $\alpha$ and $\beta$ are real numbers and satisfy $\alpha\ne\beta$ and $(\alpha,\beta)$ ∉ {$(0,1),(1,0)$}. Utilizing the monotonicity of $q_{\alpha,\beta}(t)$ on $(0,\infty)$, we derive necessary and sufficient conditions for the function $H_{a,b;c}(x)=(x+c)^{b-a} \frac{\Gamma(x+a)}{\Gamma(x+b)}$, its $q$-analogue, and ratios of the gamma or $q$-gamma functions to be logarithmically completely monotonic, where $a,b,c$ are real numbers and $x\in (-\min$ {$a,b,c$},$\infty)$.
Citation: Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231
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