How to prove the inequality provided , ? That problem was posed by Israeli mathematician nicked by himself as arqady in Russian math forum and was not answered there.I know how to prove that with Maple and don't know how to prove that without Maple. Neither nor work here. The difficulty consists in the nonlinearity both the target function and the main constraint. The first step is to linearize the main constraint and the second step is to reduce the number of variables to one.
restart; A := eval(x^(4*y)+y^(4*x), [x = sqrt(u), y = sqrt(v)]);

(1) 
B := expand(A);

(2) 
C := eval(B, u = 2v);

(3) 
It is more or less clear that the plot of F is symmetric wrt the straight line v=1. This motivates the following change of variable to obtain an even function.
F := simplify(expand(eval(C, v = z+1)), symbolic, power);

(4) 
The plots suggest the only maximim of F at z=0 and its concavity.
Student[Calculus1]:FunctionPlot(F, z = 1 .. 1);
Student[Calculus1]:FunctionPlot(diff(F, z, z), z = 1 .. 1);
As usually, numeric global solvers cannot prove certain inequalities. However, the GlobalSearch command of the DirectSearch package indicates the only local maximum of F and F''.
Digits := 25; DirectSearch:GlobalSearch(F, {z = 1 .. 1}, maximize, solutions = 3, tolerances = 10^(15)); DirectSearch:GlobalSearch(diff(F, z, z), {z = 1 .. 1}, maximize, solutions = 3, tolerances = 10^(15));

(5) 
The series command confirms a local maximum of F at z=0.
series(F, z, 6);

(6) 
The extrema command indicates only the value of F at a critical point, not outputting its position.
extrema(F, z); extrema(F, z, 's');

(7) 
solve(F = 2);

(8) 
DirectSearch:SolveEquations(F = 2, {z = 1 .. 1}, AllSolutions, solutions = 3);

(9) 
DirectSearch:SolveEquations(F = 2, {z = 1 .. 1}, AllSolutions, solutions = 3, assume = integer);

(10) 
