caa a22 4500 605508690 CHVBK 20210128100637.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s00010-014-0292-4 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0292-4 Guan Kaizhong School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People's Republic of China aut Geometrically convex solutions of a generalized gamma functional equation [Elektronische Daten] [Kaizhong Guan] In this paper we investigate geometrically convex solutions $${g : R^{+} \rightarrow R^{+}}$$ g : R + → R + to the generalized gamma functional equation with initial condition given by $$g(x + 1) = f(x)g(x) \qquad{\rm for}\,x > 0 \,{\rm and}\, g(1) = 1,\qquad\qquad\qquad(*)$$ g ( x + 1 ) = f ( x ) g ( x ) for x > 0 and g ( 1 ) = 1 , ( ∗ ) where $${f : R^{+} \rightarrow R^{+}}$$ f : R + → R + is a given function. We prove that if f satisfies some appropriate conditions, then (*) has a unique eventually geometrically convex solution g, determined by the formulae $$g(x) = \lim\limits_{n\rightarrow \infty}\frac{f(n) . . . f(1)}{f(n + x) . . . f(x)} . \Big(f(n)\Big)^{\frac{{\rm ln}(n+1+x)-{\rm ln} (n+1)}{{\rm ln}(n+1) - {\rm ln} n}}$$ g ( x ) = lim n → ∞ f ( n ) . . . f ( 1 ) f ( n + x ) . . . f ( x ) . ( f ( n ) ) ln ( n + 1 + x ) - ln ( n + 1 ) ln ( n + 1 ) - ln n $$= \lim _{n\rightarrow \infty} \frac{f(n) . . . f(1)[f(n)]^{x}}{f(n + x) . . . f(x)} \quad {\rm for}\, x > 0.$$ = lim n → ∞ f ( n ) . . . f ( 1 ) [ f ( n ) ] x f ( n + x ) . . . f ( x ) for x > 0 . Some known results in the literature are generalized and improved. Springer Basel, 2014 Convexity nationallicence Geometrical convexity nationallicence Gamma function nationallicence Functional equation nationallicence Aequationes mathematicae Springer Basel 89/4(2015-08-01), 1003-1013 0001-9054 89:4<1003 2015 89 10 https://doi.org/10.1007/s00010-014-0292-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0292-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Guan Kaizhong School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People's Republic of China aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/4(2015-08-01), 1003-1013 0001-9054 89:4<1003 2015 89 10