American Institute of Mathematical Sciences

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

Properties and applications of a function involving exponential functions

Bai-Ni Guo 1, and  Feng Qi 2,

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 $x\in (-\min$ {$a,b,c$},$\infty)$.
Keywords: necessary and sufficient condition., Monotonicity, q-gamma function, exponential function, ratio, logarithmically completely monotonic function, gamma function, logarithmic convexity.
Mathematics Subject Classification: Primary: 33B10, 33B15, 33D05; Secondary: 26A48, 26A5.
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
[1]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[2]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[3]

H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221
[4]

Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure & Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383
[5]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[6]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
[7]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
[8]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[9]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
[10]

Feng Qi, Bai-Ni Guo. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1975-1989. doi: 10.3934/cpaa.2009.8.1975
[11]

Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
[12]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51
[13]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[14]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[15]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[16]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[17]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[18]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[19]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[20]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences