After I've read the Problem 4136 from [1], and I think (I've computed it in the same way that in the solutions) that the same statement is right for the following expression where $a,b$ and $c$ are real numbers $>0$ $$b\operatorname{gd}(a)+c\operatorname{gd}(b)+a\operatorname{gd}(c)<\frac{\pi\sqrt{3}}{2}\sqrt{a^2+b^2+c^2},$$ where $\operatorname{gd}(x)$ denotes the Gudermannian function, see the Wikipedia Gudermannian function, I wodered what should be interesting or feasibles cyclic inequalities invovling particular values of the Gudermannian function and/or its inverse.
Question. Can you create a cyclic inequality, with good mathematical content, involving an expression of particular values of the Gudermannian function and/or its inverse? Many thanks.
You can take the expression involving this/these special function/s in the way that you think that the resulting inequality has good mathematical content.
References:
[1] Daniel Sitaru and Mihaly Bencze, Problem 4136, Crux Mathematicorum, Volume 43, Number 4, April 2017.