Abstract
Let psi _{n}(x)=(-1)^{n-1}psi ^{(n)}(x), where psi ^{(n)}(x) are the polygamma functions. We determine necessary and sufficient conditions for the monotonicity and convexity of the function \t\t\tF(x;α,β)=ln(exp(αψ(x+β))ψn(x))−ln(n−1)!,x>max(0,−β),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ F(x;\\alpha ,\\beta )=\\ln \\bigl(\\exp \\bigl(\\alpha \\psi (x+\\beta )\\bigr)\\psi _{n}(x) \\bigr)- \\ln (n-1)!, \\quad x>\\max (0,-\\beta ), $$\\end{document} for α and beta in mathbb{R}, where psi (x) is the psi function. Consequently, this yields not only some new inequalities for the polygamma functions, but also new star-shaped and superadditive functions involving them. In addition, we improve a well-known mean-value inequality for the polygamma functions.
Highlights
1 Introduction We recall that the logarithmic derivative of Γ (x) is called the psi or digamma function denoted by d ψ(x) = ln Γ (x) =
A world of the most fundamental properties involving the gamma, digamma, and polygamma functions can be found in some books [1–4]
It is known that these functions are all key parts of special functions
Summary
We recall that the logarithmic derivative of Γ (x) is called the psi or digamma function denoted by d. For n ∈ N are called the polygamma functions. A world of the most fundamental properties involving the gamma, digamma, and polygamma functions can be found in some books [1–4]. It is known that these functions are all key parts of special functions. They play a vital role in other areas like analysis, physics, inequality theory, and statistics. Due to their significance, they attract many scholars to explore some their useful properties. Some properties such as monotonicity, convexity, and complete monotonicity yield numerous inequalities related to these functions; see, for example, [5–28]
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