Abstract
AbstractLet us consider the logarithmic mean \(\mathcal{L,}\) the identric mean \(\mathcal{I,}\) the trigonometric means \(\mathcal{P}\) and \(\mathcal{T}\) defined by H. J. Seiffert, the hyperbolic mean \(\mathcal{N}\) defined by E. Neuman and J. Sándor, and the Gini mean \(\mathcal{J}\). The optimal estimations of these means by power means \(\mathcal{A}_{p}\) and also some of the optimal estimations by Lehmer means \(\mathcal{L}_{p}\) are known. We prove the rest of optimal estimations by Lehmer means and the optimal estimations by some other special Gini means \(\mathcal{S}_{p}\). 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).Keywordslogarithmic meanidentric meanSeiffert type meanspower meansLehmer meansspecial Gini meansinequalities of means
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