... > 0ifr s>0andJ < 0ifr s<0.Using Theorem 1.3, this completes the proof.Proof of Corollary 2.4. By the proof of Theorem 2.3, there must be J < 0ifr s<0. Notefx, y HLr, ... strictly increasing in p on 1/2, ∞ if r s>0.This proof is completed.Proof of Corollary 2.5. By the proof of Theorem 2.3, there must J < 0ifr s<0. Noticefx, y HLr, s; x, ... differences of powers,” Journal of Mathematical Analysis and Applications,vol. 131, no. 1, pp. 271–281, 1988.6 F. Qi, “Logarithmic convexity of extended mean values,” Proceedings of the American MathematicalSociety,...