Summary: Let \(g,h:[a,b] \to\mathbb{R}\) be nonnegative nondecreasing functions such that \(g\) and \(h\) have a continuous first derivative and \(g(a)=h(a)\), \(g(b)=h(b) \). Let \(p=(p_1,p_2)\) be a pair of positive real numbers \(p_1,p_2\) such that \(p_1+p_2=1\). a) If \(f:[a,b]\to\mathbb{R}\) be a nonnegative nondecreasing function, then for \(r,s1\) the inequality is reversed. b) If \(f:[a,b] \to\mathbb{R}\) is a nonnegative nonincreasing function then for \(r1>s\) the inequality is reversed. Similar results are derived for quasiarithmetic and logarithmic means.