Geometrically convex solutions of a generalized gamma functional equation
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| 001 | 605508690 | ||
| 003 | CHVBK | ||
| 005 | 20210128100637.0 | ||
| 007 | cr unu---uuuuu | ||
| 008 | 210128e20150801xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00010-014-0292-4 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0292-4 | ||
| 100 | 1 | |a Guan |D Kaizhong |u School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People's Republic of China |4 aut | |
| 245 | 1 | 0 | |a Geometrically convex solutions of a generalized gamma functional equation |h [Elektronische Daten] |c [Kaizhong Guan] |
| 520 | 3 | |a 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. | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Convexity |2 nationallicence | |
| 690 | 7 | |a Geometrical convexity |2 nationallicence | |
| 690 | 7 | |a Gamma function |2 nationallicence | |
| 690 | 7 | |a Functional equation |2 nationallicence | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/4(2015-08-01), 1003-1013 |x 0001-9054 |q 89:4<1003 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0292-4 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0292-4 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Guan |D Kaizhong |u School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People's Republic of China |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/4(2015-08-01), 1003-1013 |x 0001-9054 |q 89:4<1003 |1 2015 |2 89 |o 10 | ||