Sharp bounds for the tangent mean by weighted power mean of arithmetic and geometric means

Using the monotonicity rules for the ratios of two functions (power series), the optimal bounds for the tangent mean in terms of weighted power means of arithmetic and geometric means are obtained. The main results generalize and strengthen the known results obtained by Nowicka and Witkowski. Moreover, several remarks are presented, the sharp bounds for the hyperbolic sine mean in terms of weighted power means of arithmetic and geometric means are derived, an interesting chain of inequalities involving 12 means is established.