Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function

Abstract

Abstract In the article, we present several new monotonicity properties and bounds involving the generalized Grötzsch ring functions $\mu _{a,b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math> in the theory of Ramanujan’s generalized modular equation for $0&lt; a, b&lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math> . Our results are the variants and extensions of some previously known results.

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