| Abstrakt: |
The topic of gamma type functions and related functional equation f (x + 1) = g (x) f (x) has been seriously studied from the first half of the twentieth century till now. Regarding unique solutions of the equation the asymptotic condition lim x → ∞ g (x + w) g (x) = 1 , for each w > 0 , is considered in many serial papers including R. Webster's article (1997). Motivated by the topic of limit summability of real functions (introduced by the author in 2001), we first show that the asymptotic condition in the Webster's paper can be reduced to the convergence of the sequence g (n + 1) g (n) to 1, and then by removing the initial condition f (1) = 1 we generalize it. On the other hand, Rassias and Trif proved that if g satisfies another appropriate asymptotic condition, then the equation admits at most one solution f, which is eventually log -convex of the second order. We also sow that without changing the asymptotic condition for this case, a uniqueness theorem similar to the Webster's result is also valid for eventually log -convex solutions f of the second order. This result implies a new uniqueness theorem for the gamma function which states the log -convexity condition in the Bohr-Mollerup Theorem can be replaced by log -concavity of order two. At last, some important questions about them will be asked. [ABSTRACT FROM AUTHOR] |
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