Optimal Estimations of Seiffert-Type Means By Some Special Gini Means

摘要

Let us consider the logarithmic mean L, the identric mean I, the trigonometric means P and T defined by H. J. Seiffert, the hyperbolic mean N defined by E. Neuman and J. Sandor, and the Gini mean J. The optimal estimations of these means by power means A_p and also some of the optimal estimations by Lehmer means L_p are known. We prove the rest of optimal estimations by Lehmer means and the optimal estimations by some other special Gini means S_p. In proving some of the results we used the computer algebra system Mathematica. We believe that some parts of the proofs couldn't be done without the help of such a computer algebra system (at least by following our way of proving those results).