A convexity theorem for multiplicative functions |
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Authors: | Pierre Maréchal |
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Institution: | 1. Institut de Math??matiques de Toulouse, Universit?? Paul Sabatier, Toulouse, France
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Abstract: | We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function
g : \mathbbRm ? 0, ¥]{g : \mathbb{R}^m \rightarrow {0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β
1 + ··· + β
n
. We also provide further generalization to functions of the form (x,y1, . . . , yn)? g(x)1+af1(y1)-b1 ···fn(yn)-bn{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f
k
concave, positively homogeneous and nonnegative on their domains. |
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Keywords: | |
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