| Summary: | Abstract Let ψ n = ( − 1 ) n − 1 $\psi_{n}= ( -1 ) ^{n-1}$ ψ ( n ) $\psi^{ ( n ) }$ ( n = 0 , 1 , 2 , … $n=0,1,2,\ldots $ ), where ψ ( n ) $\psi^{ ( n ) }$ denotes the psi and polygamma functions. We prove that for n ≥ 0 $n\geq0$ and two different real numbers a and b, the function x ↦ ψ n − 1 ( ∫ a b ψ n ( x + t ) d t b − a ) − x $$ x\mapsto\psi_{n}^{-1} \biggl( \frac{\int_{a}^{b}\psi_{n}(x+t)\,dt}{b-a} \biggr) -x $$ is strictly increasing from ( − min ( a , b ) , ∞ ) $( -\min ( a,b ) ,\infty ) $ onto ( min ( a , b ) , ( a + b ) / 2 ) $( \min ( a,b ) , ( a+b ) /2 ) $ , which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.
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