What question did this study set out to answer?

The research aims to identify monotonicity rules for the ratio of two power series based on their coefficient sequences.

April 10, 2026Open Access

Monotonicity rules for the ratio of power series with applications

Key Points

The research aims to identify monotonicity rules for the ratio of two power series based on their coefficient sequences.
Analyzed the ratio of two power series by examining the sequence of coefficients
Proved the relationship between monotonicity changes in the ratio and the coefficients
Applied the findings to derive conditions for the monotonicity of specific mathematical functions.
Identified that the number of monotonicity changes in the ratio does not exceed that in the coefficient sequence
Established necessary and sufficient conditions for the monotonicity of a specific confluent hypergeometric function.

Abstract

In this paper, we present monotonicity rules for the ratio of two power series x ₊=₀^ aₖ xᵏ₊=₀^{ bₖ xᵏ} under the assumption that the sequence \aₖ/bₖ\₊ ₀ changes monotonicity twice. We also prove that the number of monotonicity changes of this function does not exceed that of the sequence \aₖ/bₖ\₊ ₀. As an application, we establish a necessary and sufficient condition for the monotonicity of a function involving the confluent hypergeometric function of the first kind, x e^-xx+1 M (a, b, cx), where a, b, c > 0.

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Authors

Zhong-Xuan Mao

Jing-Feng Tian

Journals

Applicable Analysis and Discrete Mathematics

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North China Electric Power University

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Monotonicity rules for the ratio of power series with applications

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