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In the paper, by establishing the monotonicity of some functions involvingthe sine and cosine functions, the authors provide concise proofs of some knowninequalities and find some new sharp inequalities involving the Seiffert,contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-squaremeans of two positive real numbers $a$ and $b$ with $a\ne b$.

Some sharp inequalities involving Seiffert and other means and their concise proofs