Optimal bounds of exponential type for arithmetic mean by Seiffert-like mean and centroidal mean

In this paper, optimal bounds for arithmetic mean in terms of hyperbolic sine mean and centroidal mean, the tangent mean and centroidal mean in exponential type are established using the monotone form of L’Hospital’s rule and the criterions for the monotonicity of the quotient of power series. Based on two basic conclusions, we carefully compare them with the existing inequalities involving the four means mentioned above, and obtain a new refined inequality chain.