Multiplicative properties of real functions with applications to classical functions

Abstract

From the characterisation of geometrically convex and geometrically concave functions defined on (0, A] or [A, ∞) with A > 0, by means of their multiplicative conditions, we obtain unified proofs of some known and new inequalities. Functions of class C 2 and strictly increasing on (a, b) fulfil some kind of supermultiplicativity and superadditivity. We have obtained a new constant determining the intervals of sub-and supermultiplicativity for the log function.Mathematics Subject Classification (1991). 26A09, 26A48, 26A51, 26D05, 26D15, 33B15.

Product inequalities for norms of linear factors

“…The condition that log(g(r)) is a convex function of log(r) is sometimes stated as g is multiplicatively convex [9] or g is geometrically convex [3]. The next theorem shows that such functions arise in a natural way in complex analysis.…”

“…We will also need the following result which is a multiplicative version of Petrović's inequality [10] due to Finol and Wójtowicz [3]. For the sake of completeness, we provide a direct proof.…”

Who's Afraid of Selection Bias? Robust Inference in the Presence of Competitive Selection

“…Thus, the question of selection dominance reduces to the question of what restrictions that must be imposed on the transformation u = F • G −1 in order to ensure that (17) holds. The following result, which is essentially specialized and simplified statement of Theorem 1 in Finol and Wójtowicz (2000), will prove to be of great assistance in answering this question.…”

On a Convexity Theorem of Ruskai and Werner and Related Results

“…REMARK 3. Comments on the relevance of sub-and supermultiplicative functions in various fields as well as references on this subject can be found in [6]. REMARK 4.…”