- Email: [email protected]
Some inequalities for the gamma function
Applied Mathematics and Computation 218 (2011) 4349–4352
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some inequalities for the gamma function Edward Neuman Department of Mathematics, Mailcode 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901, USA
a r t i c l e
i n f o
a b s t r a c t Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Gamma function Digamma function Logarithmically convex functions Logarithmically concave functions Inequalities
1. Introduction and notation One of the most important special function is Euler’s gamma function
CðxÞ ¼
Z
1
et tx1 dt;
0
ðx > 0Þ. This function is of great importance in probability theory, mathematical statistics, physics, just to mention a few areas only. Another special function used in this paper is the logarithmic derivative W of the gamma function
WðxÞ ¼
C0 ðxÞ : CðxÞ
ð1:1Þ
It is well known that Euler’s function is strictly log-convex on R> , i.e.,
Cðux þ ð1 uÞyÞ < ½CðxÞu ½CðyÞ1u ; holds for 0 < u < 1 and all x; y 2 R> ðx – yÞ. See, e.g., [2,7]. The goal of this paper is to establish inequalities involving ratios and products of the gamma function. These results are presented in Section 2. Numerous inequalities for the gamma function are known in mathematical literature. The interested reader is referred to [1–5,7,8,11,12] and to the references therein. 2. Main results In what follows the symbol D will stand for the subinterval of the number line R while R> will be used to denote the positive number line. Also, we will assume that a function f : D ! R> is log-convex. We define
/ /ða; b; xÞ ¼
1 f ða þ xÞ ab ; f ðb þ xÞ
a þ x;
b þ x 2 D;
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.010
a – b:
ð2:1Þ
4350
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
Proposition 2.1. /ða; b; xÞ increases with an increase in either a or b. Proof. Let
U :¼ ln / ¼
ln f ða þ xÞ ln f ðb þ xÞ : ab
ð2:2Þ
Since U is the first order divided difference of a convex function, it is an increasing function in both a þ x and b þ x. This in turn implies / has the desired monotonicity property in variables a and b. h Corollary 2.1. If f is continuously differentiable on D, then the following inequalities
f 0 ðb þ xÞ f ða þ xÞ f 0 ða þ xÞ 6 ln 6 ða bÞ ða þ x; b þ x 2 D; a – bÞ; f ðb þ xÞ f ðb þ xÞ f ða þ xÞ Cða þ xÞ ða bÞWðb þ xÞ 6 ln 6 ða bÞWða þ xÞ ða þ x; b þ x 2 R> Þ Cðb þ xÞ
ða bÞ
ð2:3Þ ð2:4Þ
and
C 1þx 1 1 1 6 2xW x 6 ln 21 þ x jxj < 2xW 2 2 2 C 2x
ð2:5Þ
hold true. Proof. It follows from (2.1) and Proposition 2.1 that
ða bÞ2
@U f 0 ða þ xÞ f ða þ xÞ ln P0 ¼ ða bÞ f ða þ xÞ f ðb þ xÞ @a
ða bÞ2
@U f 0 ðb þ xÞ f ða þ xÞ ¼ ða bÞ þ ln P 0: @b f ðb þ xÞ f ðb þ xÞ
and
Hence the inequalities (2.3) follow. For the proof of (2.4) we let in (2.3) f ðtÞ ¼ CðtÞ ðt > 0Þ and next utilize (1.1). Inequalities (2.5) are obtained from (2.4) by letting a ¼ x; b ¼ x, and x ¼ 12. h Second inequality in (2.4) provides a generalization of one of Gautschi inequalities for the gamma function (see [3]). For more results about properties and bounds for the ratios of gamma function the interested reader is referred to [3,5,10,11]. Corollary 2.2. The functions
1 xy
Cð1 þ xÞ Cð1 þ yÞ
1
ðx; y > 1; x – yÞ and ½Cð1 þ xÞx ðx > 1; x – 0Þ are increasing in variables x and y. Proof. The first statement is a consequence of Proposition 2.1. For, we let f ðtÞ ¼ CðtÞ; a ¼ x; b ¼ y, and x ¼ 1. Monotonicity of the second function follows from monotonicity of the first one by letting y ¼ 0. h The following lemma will be used in the sequel. Lemma 2.1. Let g : Rþ ! R and let xi > 0; 1 6 i 6 n. If the function gðxÞ=x is increasing on R> , then n X
gðxi Þ 6 gðrÞ;
i¼1
where
r ¼ x1 þ þ x n : Inequality (2.6) is reversed if gðxÞ=x is decreasing on R> . Proof. Assume that the function gðxÞ=x is increasing on R> . Taking into account that xi 6 r for 1 6 i 6 n we obtain
ð2:6Þ
4351
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
gðrÞ ¼
n X
xi
gðrÞ
i¼1
r
P
n X i¼1
xi
n gðxi Þ X ¼ gðxi Þ: xi i¼1
Case when gðxÞ=x is decreasing on R> can be treated in an analogous manner. We omit further details. h Proposition 2.2. The following inequalities
h
C 1þ
r in n
6
n Y
Cð1 þ xi Þ 6 Cð1 þ rÞðxi > 1; 1 6 i 6 nÞ
ð2:7Þ
i¼1
and 1
½Cð1 þ rÞr 6
n X 1 ½Cð1 þ xi Þxi ðxi > 0; 1 6 i 6 nÞ
ð2:8Þ
i¼1
are valid. Proof. The first inequality in (2.7) is a consequence of the logarithmic convexity of the gamma function. In order to establish 1 the second inequality in (2.7) we utilize a fact that the function ½Cð1 þ xÞx is increasing for x > 1 (see Corollary 2.2). This in turn implies that its logarithm 1
ln ½Cð1 þ xÞx ¼
ln Cð1 þ xÞ x
is also increasing for x > 1. Application of Lemma 2.1 with gðxÞ ¼ ln Cð1 þ xÞ gives n X
ln Cð1 þ xi Þ 6 ln Cð1 þ rÞ:
i¼1 1
Hence the assertion follows. For the proof of the inequality (2.8) we will use Lemma 2.1 with gðxÞ ¼ ½Cð1 þ xÞx ðx > 0Þ. Kershaw and Laforgia [6] have proven that the function gðxÞ=x is strictly decreasing on R> . The assertion now follows. h Let gðxÞ be the same as in the proof of Proposition 2.2. It is known that gðxÞ is log-concave for x > 1 (see [4]). This yields a lower bound for the first member of (2.7) n h Y r r in ½Cð1 þ xi Þnxi 6 C 1 þ : n i¼1
It is worth mentioning that gðxÞ is concave for x P 7 (see [9]). Corollary 2.3. Let x P 0 and let n ¼ 1; 2; . . . . Then
nnx 6
½Cð1 þ xÞn 61 Cð1 þ nxÞ
ð2:9Þ
and
1 1 p ffiffiffi 6 6 1: C 1 þ n n n
ð2:10Þ
Proof. Combining the last two members of (2.7) with the inequality (2.8) we obtain n Y
( )r n X 1 xi Cð1 þ xi Þ 6 Cð1 þ rÞ 6 ½Cð1 þ xi Þ ðxi > 0; 1 6 i 6 nÞ:
i¼1
i¼1
Letting above x1 ¼ ¼ xn ¼ x > 0, we obtain
½Cð1 þ xÞn 6 Cð1 þ nxÞ 6 nnx ½Cð1 þ xÞn : Hence the inequalities (2.9) follow. For the proof of (2.10) we put x ¼ 1n in (2.9). h Alisina and Tomás [1] have employed a geometric argument to prove that for any 0 6 x 6 1 and positive integer n the following inequalities
1 ½Cð1 þ xÞn 6 61 n! Cð1 þ nxÞ
ð2:11Þ
4352
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
are valid. An easy induction, with use of the inequality ðn þ 1Þn1 6 nn , shows that nnx < n! holds for 0 < x 6 12 and n ¼ 1; 2; . . .. Thus the first inequality in (2.9) is tighter than the corresponding inequality in (2.11) provided x 2 ð0; 12. Generalizations of (2.11) are obtained in [12,8]. Acknowledgments This author would like to express his gratitude to Professor József Sándor for helpful suggestions and remarks which had an impact on the present version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
C. Alisina, M.S. Tomás, A geometrical proof of a new inequality for the gamma function, J. Ineq. Pure Appl. Math. 6 (2) (2005) (Art. 48). B.C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959) 77–81. P.J. Grabner, R. Thichy, U.T. Zimmermann, Inequalities for the gamma function with applications to permanents, Discrete Math. 154 (1996) 53–62. D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983) 607–611. D. Kershaw, A. Laforgia, Monotonicity results for the gamma function, Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Nature 119 (1985) 127–133. D.S. Mitrinowic´, Analytic Inequalities, Springer-Verlag, Berlin, 1970. E. Neuman, Inqualities involving a logarithmically convex function and their applications to special functions, J. Ineq. Pure Appl. Math. 7 (1) (2006) (Art. 16). J. Sándor, Sur la fonction gamma, Publ. C.R. Math. Pures Neuchatel, Serie 1 (21) (1989) 4–7. J. Sándor, On the gamma function III, Publ. C.R. Math. Pures Neuchatel, Serie 2 (19) (2001) 33–40. J. Sándor, On certain inequalities for the ratios of the gamma function, Octogon Math. Mag. 12 (2004) 1052–1059. J. Sándor, A note on certain inequalities for the gamma function, J. Ineq. Pure Appl. Math. 6 (3) (2005) (Art. 61).
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some inequalities for the gamma function Edward Neuman Department of Mathematics, Mailcode 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901, USA
a r t i c l e
i n f o
a b s t r a c t Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Gamma function Digamma function Logarithmically convex functions Logarithmically concave functions Inequalities
1. Introduction and notation One of the most important special function is Euler’s gamma function
CðxÞ ¼
Z
1
et tx1 dt;
0
ðx > 0Þ. This function is of great importance in probability theory, mathematical statistics, physics, just to mention a few areas only. Another special function used in this paper is the logarithmic derivative W of the gamma function
WðxÞ ¼
C0 ðxÞ : CðxÞ
ð1:1Þ
It is well known that Euler’s function is strictly log-convex on R> , i.e.,
Cðux þ ð1 uÞyÞ < ½CðxÞu ½CðyÞ1u ; holds for 0 < u < 1 and all x; y 2 R> ðx – yÞ. See, e.g., [2,7]. The goal of this paper is to establish inequalities involving ratios and products of the gamma function. These results are presented in Section 2. Numerous inequalities for the gamma function are known in mathematical literature. The interested reader is referred to [1–5,7,8,11,12] and to the references therein. 2. Main results In what follows the symbol D will stand for the subinterval of the number line R while R> will be used to denote the positive number line. Also, we will assume that a function f : D ! R> is log-convex. We define
/ /ða; b; xÞ ¼
1 f ða þ xÞ ab ; f ðb þ xÞ
a þ x;
b þ x 2 D;
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.010
a – b:
ð2:1Þ
4350
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
Proposition 2.1. /ða; b; xÞ increases with an increase in either a or b. Proof. Let
U :¼ ln / ¼
ln f ða þ xÞ ln f ðb þ xÞ : ab
ð2:2Þ
Since U is the first order divided difference of a convex function, it is an increasing function in both a þ x and b þ x. This in turn implies / has the desired monotonicity property in variables a and b. h Corollary 2.1. If f is continuously differentiable on D, then the following inequalities
f 0 ðb þ xÞ f ða þ xÞ f 0 ða þ xÞ 6 ln 6 ða bÞ ða þ x; b þ x 2 D; a – bÞ; f ðb þ xÞ f ðb þ xÞ f ða þ xÞ Cða þ xÞ ða bÞWðb þ xÞ 6 ln 6 ða bÞWða þ xÞ ða þ x; b þ x 2 R> Þ Cðb þ xÞ
ða bÞ
ð2:3Þ ð2:4Þ
and
C 1þx 1 1 1 6 2xW x 6 ln 21 þ x jxj < 2xW 2 2 2 C 2x
ð2:5Þ
hold true. Proof. It follows from (2.1) and Proposition 2.1 that
ða bÞ2
@U f 0 ða þ xÞ f ða þ xÞ ln P0 ¼ ða bÞ f ða þ xÞ f ðb þ xÞ @a
ða bÞ2
@U f 0 ðb þ xÞ f ða þ xÞ ¼ ða bÞ þ ln P 0: @b f ðb þ xÞ f ðb þ xÞ
and
Hence the inequalities (2.3) follow. For the proof of (2.4) we let in (2.3) f ðtÞ ¼ CðtÞ ðt > 0Þ and next utilize (1.1). Inequalities (2.5) are obtained from (2.4) by letting a ¼ x; b ¼ x, and x ¼ 12. h Second inequality in (2.4) provides a generalization of one of Gautschi inequalities for the gamma function (see [3]). For more results about properties and bounds for the ratios of gamma function the interested reader is referred to [3,5,10,11]. Corollary 2.2. The functions
1 xy
Cð1 þ xÞ Cð1 þ yÞ
1
ðx; y > 1; x – yÞ and ½Cð1 þ xÞx ðx > 1; x – 0Þ are increasing in variables x and y. Proof. The first statement is a consequence of Proposition 2.1. For, we let f ðtÞ ¼ CðtÞ; a ¼ x; b ¼ y, and x ¼ 1. Monotonicity of the second function follows from monotonicity of the first one by letting y ¼ 0. h The following lemma will be used in the sequel. Lemma 2.1. Let g : Rþ ! R and let xi > 0; 1 6 i 6 n. If the function gðxÞ=x is increasing on R> , then n X
gðxi Þ 6 gðrÞ;
i¼1
where
r ¼ x1 þ þ x n : Inequality (2.6) is reversed if gðxÞ=x is decreasing on R> . Proof. Assume that the function gðxÞ=x is increasing on R> . Taking into account that xi 6 r for 1 6 i 6 n we obtain
ð2:6Þ
4351
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
gðrÞ ¼
n X
xi
gðrÞ
i¼1
r
P
n X i¼1
xi
n gðxi Þ X ¼ gðxi Þ: xi i¼1
Case when gðxÞ=x is decreasing on R> can be treated in an analogous manner. We omit further details. h Proposition 2.2. The following inequalities
h
C 1þ
r in n
6
n Y
Cð1 þ xi Þ 6 Cð1 þ rÞðxi > 1; 1 6 i 6 nÞ
ð2:7Þ
i¼1
and 1
½Cð1 þ rÞr 6
n X 1 ½Cð1 þ xi Þxi ðxi > 0; 1 6 i 6 nÞ
ð2:8Þ
i¼1
are valid. Proof. The first inequality in (2.7) is a consequence of the logarithmic convexity of the gamma function. In order to establish 1 the second inequality in (2.7) we utilize a fact that the function ½Cð1 þ xÞx is increasing for x > 1 (see Corollary 2.2). This in turn implies that its logarithm 1
ln ½Cð1 þ xÞx ¼
ln Cð1 þ xÞ x
is also increasing for x > 1. Application of Lemma 2.1 with gðxÞ ¼ ln Cð1 þ xÞ gives n X
ln Cð1 þ xi Þ 6 ln Cð1 þ rÞ:
i¼1 1
Hence the assertion follows. For the proof of the inequality (2.8) we will use Lemma 2.1 with gðxÞ ¼ ½Cð1 þ xÞx ðx > 0Þ. Kershaw and Laforgia [6] have proven that the function gðxÞ=x is strictly decreasing on R> . The assertion now follows. h Let gðxÞ be the same as in the proof of Proposition 2.2. It is known that gðxÞ is log-concave for x > 1 (see [4]). This yields a lower bound for the first member of (2.7) n h Y r r in ½Cð1 þ xi Þnxi 6 C 1 þ : n i¼1
It is worth mentioning that gðxÞ is concave for x P 7 (see [9]). Corollary 2.3. Let x P 0 and let n ¼ 1; 2; . . . . Then
nnx 6
½Cð1 þ xÞn 61 Cð1 þ nxÞ
ð2:9Þ
and
1 1 p ffiffiffi 6 6 1: C 1 þ n n n
ð2:10Þ
Proof. Combining the last two members of (2.7) with the inequality (2.8) we obtain n Y
( )r n X 1 xi Cð1 þ xi Þ 6 Cð1 þ rÞ 6 ½Cð1 þ xi Þ ðxi > 0; 1 6 i 6 nÞ:
i¼1
i¼1
Letting above x1 ¼ ¼ xn ¼ x > 0, we obtain
½Cð1 þ xÞn 6 Cð1 þ nxÞ 6 nnx ½Cð1 þ xÞn : Hence the inequalities (2.9) follow. For the proof of (2.10) we put x ¼ 1n in (2.9). h Alisina and Tomás [1] have employed a geometric argument to prove that for any 0 6 x 6 1 and positive integer n the following inequalities
1 ½Cð1 þ xÞn 6 61 n! Cð1 þ nxÞ
ð2:11Þ
4352
E. Neuman / Applied Mathematics and Computation 218 (2011) 4349–4352
are valid. An easy induction, with use of the inequality ðn þ 1Þn1 6 nn , shows that nnx < n! holds for 0 < x 6 12 and n ¼ 1; 2; . . .. Thus the first inequality in (2.9) is tighter than the corresponding inequality in (2.11) provided x 2 ð0; 12. Generalizations of (2.11) are obtained in [12,8]. Acknowledgments This author would like to express his gratitude to Professor József Sándor for helpful suggestions and remarks which had an impact on the present version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
C. Alisina, M.S. Tomás, A geometrical proof of a new inequality for the gamma function, J. Ineq. Pure Appl. Math. 6 (2) (2005) (Art. 48). B.C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959) 77–81. P.J. Grabner, R. Thichy, U.T. Zimmermann, Inequalities for the gamma function with applications to permanents, Discrete Math. 154 (1996) 53–62. D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983) 607–611. D. Kershaw, A. Laforgia, Monotonicity results for the gamma function, Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Nature 119 (1985) 127–133. D.S. Mitrinowic´, Analytic Inequalities, Springer-Verlag, Berlin, 1970. E. Neuman, Inqualities involving a logarithmically convex function and their applications to special functions, J. Ineq. Pure Appl. Math. 7 (1) (2006) (Art. 16). J. Sándor, Sur la fonction gamma, Publ. C.R. Math. Pures Neuchatel, Serie 1 (21) (1989) 4–7. J. Sándor, On the gamma function III, Publ. C.R. Math. Pures Neuchatel, Serie 2 (19) (2001) 33–40. J. Sándor, On certain inequalities for the ratios of the gamma function, Octogon Math. Mag. 12 (2004) 1052–1059. J. Sándor, A note on certain inequalities for the gamma function, J. Ineq. Pure Appl. Math. 6 (3) (2005) (Art. 61).