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arXiv · 1301.3264

Geometric convexity of the generalized sine and the generalized hyperbolic sine

Wei-Dong Jiang·Feng Qi
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Abstract

In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors verify a conjecture posed in the paper "B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at \url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".

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Wei-Dong Jiang, Feng Qi. 2013-01-15. Geometric convexity of the generalized sine and the generalized hyperbolic sine. https://doi.org/10.1016/j.jat.2013.06.005

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