Geometric convexity of the generalized sine and the generalized hyperbolic sine

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}".

5 pages

References in corpus (2)

• Inequalities for generalized trigonometric and hyperbolic sine functions
• Inequalities for the generalized trigonometric and hyperbolic functions

Cited by in corpus (5)

• Turán type inequalities for generalized inverse trigonometric functions
• On generalized complete $(p,q)$-elliptic integrals
• Logarithmic mean inequality for generalized trigonometric and hyperbolic functions
• On the conjecture of generalized trigonometric and hyperbolic functions
• Convexity properties of generalized trigonometric and hyperbolic functions