Abstract
Complete Monotonicity Properties of a Function Involving the Polygamma Function
Highlights
Introduction and PreliminariesThe classical Gamma function, which is an extension of the factorial notation to noninteger values is usually defined as ∞Γ(x) = tx−1e−t dt, x > 0, and satisfying the basic propertyΓ(x + 1) = xΓ(x), x > 0.Its logarithmic derivative, which is called the Psi or digamma function is defined as ψ(x) d dx ln Γ(x) −γ∞ 0 e−t − e−xt 1 − e−t dt, x > 0, (1)
We study completete monotonicity properties of the function fa,k(x)
In [6], Qiu and Vuorinen established among other things that the function h1 = ψ x is strictly decreasing and convex on (0, ∞)
Summary
We study completete monotonicity properties of the function fa,k(x) We deduce some inequalities involving the polygamma functions. Its logarithmic derivative, which is called the Psi or digamma function is defined as ∞ 0 tne−xt 1 − e−t dt, x > 0, (2) X > 0, k=0 satisfying the functional equation [1, p.
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