Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→sinp⁡(y)p→sinp⁡(y) and p→cosp⁡(y)p→cosp⁡(y) are log-concave on the appropriate intervals while p→tanp⁡(y)p→tanp⁡(y) is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].