PROPERTIES AND APPLICATIONS OF A FUNCTION INVOLVING EXPONENTIAL FUNCTIONS

In the present paper, we give necessary and sufficient conditions for the elementary function q(alpha,beta)(t) = {(e-alpha t-e-beta t/1-e-t)(beta - alpha,), (t = 0) (t not equal 0) to be monotonic or logarithmically convex on (-infinity, infinity), (-infinity, 0) or (0, infinity) respectively, where a and beta are real numbers and satisfy alpha not equal beta and (alpha, beta) is not an element of {(0,1), (1,0)}. Utilizing the monotonicity of q(alpha,beta)(t) on (0, infinity), we derive necessary and sufficient conditions for the function H-a,H-b;c(x) = (x + c)(b-a) 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 is an element of ( - min {a, b, c}, infinity).