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Monotonicity theorems and inequalities for the Hübner function with applications. (English) Zbl 1459.33014

The authors present sharp bounds for the Hübner function, with applications to the theories of quasiconformal maps and Ramanujan’s modular equations. They prove the monotonicity and concavity-convexity properties of certain combinations of the Hübner function with elementary functions. Besides those bounds, they improve the associated logarithmic inequalities, the explicit quasiconformal Schwarz lemma and the estimates of the solutions to Ramanujan’s classical modular equations.

MSC:

33E30 Other functions coming from differential, difference and integral equations
26A48 Monotonic functions, generalizations
30C62 Quasiconformal mappings in the complex plane

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