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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Zhong-Xuan Mao
Jing-Feng Tian
Applicable Analysis and Discrete Mathematics
North China Electric Power University
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Mao et al. (Thu,) studied this question.
www.synapsesocial.com/papers/69d893406c1944d70ce044e8 — DOI: https://doi.org/10.2298/aadm250601011m
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