Abstract
Using symmetric forms\(M_{k_1 ,...,k_p } : = (1/A(n))\,\sum\limits_{v_1 ,...,v_p } {a_{v_1 }^{k_1 } ...a_{v_p }^{k_p } } \) A(n)=number of terms of the sum,a ν>0,k i≠0,i=1,...,n) the meansm k 1,...,kp:=(Mk 1,...,kp1/(k1+...+kp)(k1+...+kp≠0) are formed and investigated as to monotonicity under the hypothesis that the exponentsk 1,...,k p are certain linear functions of only one parameterk(k i =λ i k+β 1,λ i >0,β 1+...β p =0). (The means\(m_{\lambda _1 k_1 ,...,\lambda _1 k_p } \), e. g., are increasing ifk is increasing.) The proofs are elementary and use the known method of positive logarithmically convex (or concave) sequences and certain generalizations of Muirhead's theorem.
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Beck, E. Über Mittelwerte aus Potenzproduktsummen. Monatshefte für Mathematik 83, 177–189 (1977). https://doi.org/10.1007/BF01541634
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DOI: https://doi.org/10.1007/BF01541634