On the special form of integral convolution type inequality due to Walter and Weckesser

Walter and Weckesser's result (Aequationes Math 46: 212-219, 1993), extending the Bushell-Okrasi ' nski convolution type inequality (Bushell and Okrasi ' nski in J Lond Math Soc (2) 41: 503-510, 1990), gave some general conditions on the functions k : [ 0, d). R and g : [ 0,8). R under which, for every increasing function f : [ 0, d). [ 0,8), the inequality x 0 k (x -s) g (f (s)) ds = g x 0 f (s) ds , x. (0, d), is satisfied. Applying the result on a simultaneous system of functional inequalities, we prove that if d > 1, then, in general, both k and g must be power functions.