Gheorghe Toader

A Nice Separation of Some Seiffert-Type Means by Power Means

Seiffert has defined two well-known trigonometric means denoted by 𝒫 and 𝒯. In a similar way it was defined by Carlson the logarithmic mean ℒ as a hyperbolic mean. Neuman and Sandor completed the list of such means by another hyperbolic mean ℳ. There are more known inequalities between the means 𝒫,𝒯, and ℒ and some power means 𝒜𝑝. We add to these inequalities two new results obtaining the following nice chain of inequalities 𝒜0<ℒ<𝒜1/2<𝒫<𝒜1<ℳ<𝒜3/2<𝒯<𝒜2, where the power means are evenly spaced with respect to their order.

On some exponential means. Part II

We prove some new inequalities involving an exponential mean, its complementary, and some means derived from known means by applying the exp-log method.

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

Let 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. Sandor, 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).

Invariance in the Family of Weighted Gini Means

Given two means M and N, the mean P is called \((M,N)\)-invariant if \(P(M,\) \(N)=P.\) At the same time the mean N is called complementary to M with respect to P. We use the method of series expansion of means to determine the complementary with respect to a weighted Gini mean. The invariance in the family of weighted Gini means is also studied. The computer algebra Maple was used for solving some complicated systems of equations.

Invariance in Some Families of Means

A mean P is (M, N)-invariant if P(M, N) = P. In the same time the mean N is called complementary to M with respect to P. For the determination of complementaries, three methods have been used: the direct calculation, the methods of functional equations, and the series expansion of means. In the current paper we consider the method of series expansion of means to study the invariance in the family of extended logarithmic means.