Monotonicity and positivity of several functions involving ratios and products of polygamma functions

Abstract

Abstract Let ψ ( x ) $\psi (x)$ denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function Γ ( x ) $\Gamma (x)$ . In the paper, the authors discover the monotonic properties of the functions ψ ( n ) ( x ) x ψ ( n + 1 ) ( x ) and ψ ( n ) ( x ) ψ ( n ) ( 1 x ) $$ \frac{\psi ^{(n)}(x)}{x\psi ^{(n+1)}(x)} \quad \text{and}\quad \psi ^{(n)}(x) \psi ^{(n)}\biggl(\frac{1}{x}\biggr) $$ for n ≥ 0 $n\ge 0$ on ( 0 , ∞ ) $(0,\infty )$ . With the aid of these monotonic properties, the authors confirm the positivity of the function ψ ( x ) + x ψ ′ ( x ) − ψ ( x ) ψ ( 1 x ) $$ \psi (x)+x\psi '(x)-\psi (x)\psi \biggl(\frac{1}{x}\biggr) $$ on ( 0 , ∞ ) $(0,\infty )$ . The authors also pose five open problems, generalizing the latter two of the three functions mentioned above and their related conclusions.

Keywords

• Monotonicity
• Positivity
• Ratio
• Product
• Gamma function
• Digamma function