Extension of Karamata inequality for generalized inverse trigonometric functions

Discussing Ramanujan's Question 294, Karamata established the inequality log x /x-1 <= 1+3 root x / x + 3 root x, (x > 0, x not equal 1), (*) which is the cornerstone of this paper. We generalize the above inequality transforming into terms of arctan and artanh. Moreover, we expand the established result to the class of generalized inverse p-trigonometric arctanp and to hyperbolic artanh(p) functions.