Generalization of Some Inequalities for the Ratio of Gamma Functions

Transcription

1 In. Journal of Mah. Analysis, Vol. 8, 2014, no. 18, HIKARI Ld, hp://dx.doi.org/ /ijma Generalizaion of Some Inequaliies for he Raio of Gamma Funcions K. Nanomah and M. M. Iddrisu Deparmen of Mahemaics Universiy for Developmen Sudies Navrongo Campus, P. O. Box 24 Navrongo, UE/R, Ghana E. Prempeh Deparmen of Mahemaics Kwame Nrumah Universiy of Science and Technology Kumasi, Ghana Copyrigh c 2014 K. Nanomah, M. M. Iddrisu and E. Prempeh. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original wor is properly cied. Absrac We presen some monoonic funcions and some generalized inequaliies involving he raios of analogues of he Gamma funcion. Mahemaics Subjec Classificaion: 33B15, 26A48 Keywords: Gamma Funcion, p-analogue, q-analogue, -analogue, Inequaliy 1 Inroducion The classical Euler s Gamma funcion Γ() is commonly defined as Γ() = 0 e x x 1 dx, > 0. (1)

2 896 K. Nanomah, M. M. Iddrisu and E. Prempeh The p-digamma funcion ψ p (), q-digamma funcion ψ q () and -digamma funcion ψ () are respecively defined as follows. ψ p () = d d ln(γ p()) = Γ p(), > 0. (2) Γ p () where Γ p () is he p-analogue of he Gamma funcion defined by (see [2], [3]) Γ p () = p!p ( +1)...( + p) = p (1 + )...(1 + p N, > 0, (3) ), 1 p ψ q () = d d ln(γ q()) = Γ q () Γ q (), > 0 (4) where Γ q () is he q-analogue of he Gamma funcion defined by (see [4]) and Γ q () =(1 q) 1 1 q n, q (0, 1), > 0, (5) 1 q+n ψ () = d d ln(γ ()) = Γ () Γ (), > 0 (6) where Γ () is he -analogue of he Gamma funcion defined by (see [1], [5]) Γ () = 0 e x x 1 dx, > 0, > 0. (7) In a recen paper [6], Nanomah and Iddrisu proved ha he following double inequaliies hold: and 0 < Γ ln γ (α + ) Γ p (α + ) < e( 1)( ) Γ (α +1), > 0, p N (8) p 1 Γ p (α +1) 0 < Γ ln γ (α + ) Γ q (α + ) < e( )( ) Γ (α +1), > 0, q (0, 1) (9) (1 q) 1 Γ q (α +1) for (0, 1) and for a posiive real number α. Our objecive in his paper is o esablish some generalizaions of he inequaliies (8) and (9).

3 Generalizaion of some inequaliies Preliminary Resuls The following auxiliary resuls are crucial o he main resuls of he paper. Lemma 2.1. The funcions ψ p (), ψ q () and ψ () as defined above have he following series represenaions. ψ p () =lnp p ψ q () = ln(1 q)+lnq 1, p N, > 0 (10) n + q +n, q (0, 1), > 0 (11) 1 q+n ψ () = ln γ 1 +, > 0, > 0. (12) n(n + ) where γ is he Euler-Mascheroni s consan. Proof. See [3], [5] and [6] and he references herein. Lemma 2.2. Le a>0, b>0 and >0. Then, a( ln γ )+bln p + a + aψ () bψ p () > 0. Proof. Using he series represenaions in equaions (10) and (12) we have, a( ln γ )+bln p + a + aψ () bψ p () = a p n(n + ) + b 1 (n + ) > 0. Lemma 2.3. Le a>0, b>0 and α + β > 0. Then a( ln γ )+bln p + a + aψ (α + β) bψ p (α + β) > 0 Proof. This follows direcly from Lemma 2.2. Lemma 2.4. Le a>0, b>0 and >0. Then, a( ln γ )+bln(1 q)+ a + aψ () bψ q () > 0

4 898 K. Nanomah, M. M. Iddrisu and E. Prempeh Proof. Using he series represenaions in equaions (11) and (12) we have, a( ln γ )+bln(1 q)+ a + aψ () bψ q () = a n(n + ) b ln q q +n 1 q +n > 0 Lemma 2.5. Le a>0, b>0 and α + β > 0. Then, a( ln γ )+bln(1 q)+ a + aψ (α + β) bψ p (α + β) > 0 Proof. This follows direcly from Lemma Main Resuls We now sae and prove he resuls of his paper. Theorem 3.1. Define a funcion Λ by Λ() = aβ e aβ( ln γ ) Γ (α + β) a, (0, ), >0, p N. (13) p bβ Γ p (α + β) b where a, b, α, β are posiive real numbers. Then Λ is increasing on (0, ) and he inequaliy 0 < Γ (α + β) a ln γ Γ p (α + β) < eaβ( 1)( ) Γ (α + β) a (14) b aβ p bβ( 1) Γ p (α + β) b holds for every (0, 1). Proof. Le g() = ln Λ() for every (0, ). Then, g() =ln aβ e aβ( ln γ ) Γ (α + β) a p bβ Γ p (α + β) b = aβ( ln γ )+bβ ln p + aβ ln + a ln Γ (α + β) b ln Γ p (α + β) Then, g () = aβ( ln γ )+bβ ln p + aβ + aβψ (α + β) bβψ p (α + β) [ = β a( ln γ )+bln p + a ] + aψ (α + β) bψ p (α + β) > 0

5 Generalizaion of some inequaliies 899 as a resul of Lemma 2.3. This proves ha g is increasing on (0, ). Hence Λ is increasing on (0, ). Thus, for every (0, 1) we have yielding Λ(0) < Λ() < Λ(1), 0 < Γ (α + β) a ln γ Γ p (α + β) < eaβ( 1)( ) Γ (α + β) a. b aβ p bβ( 1) Γ p (α + β) b Corollary 3.2. If [1, ), hen he following inequaliy holds. e ) Γ (α + β) a aβ p bβ( 1) Γ p (α + β) b aβ( 1)( ln γ Γ (α + β) a Γ p (α + β) b Proof. If [1, ), hen we have Λ(1) Λ() yielding he resul. Theorem 3.3. Define a funcion Υ by Υ() = aβ e aβ( ln γ ) Γ (α + β) a, (0, ), >0, q (0, 1). (15) (1 q) bβ Γ q (α + β) b where a, b, α, β are posiive real numbers. Then Υ is increasing on (0, ) and he inequaliy holds for every (0, 1). 0 < Γ (α + β) a Γ q (α + β) b < Proof. Le h() = ln Υ() for every (0, ). Then, h() =ln aβ e aβ( ln γ ) Γ (α + β) a (1 q) bβ Γ q (α + β) b ln γ eaβ( 1)( ) Γ (α + β) a (16) aβ (1 q) bβ(1 ) Γ q (α + β) b = aβ( ln γ ) bβ ln(1 q)+aβ ln + a ln Γ (α + β) b ln Γ q (α + β) Then, h () = aβ( ln γ ) bβ ln(1 q)+ aβ + aβψ (α + β) bβψ q (α + β) [ = β a( ln γ ) b ln(1 q)+ a ] + aψ (α + β) bψ p (α + β) > 0

6 900 K. Nanomah, M. M. Iddrisu and E. Prempeh as a resul of Lemma 2.5. This proves ha h is increasing on (0, ). Hence Υ is increasing on (0, ) and for every (0, 1) we have Υ(0) < Υ() < Υ(1) yielding 0 < Γ (α + β) a Γ q (α + β) b < ln γ eaβ( 1)( ) Γ (α + β) a aβ (1 q) bβ(1 ) Γ q (α + β). b Corollary 3.4. If [1, ), hen he following inequaliy holds. e ) Γ (α + β) a aβ (1 q) bβ(1 ) Γ q (α + β) Γ (α + β) a b Γ q (α + β) b aβ( 1)( ln γ Proof. If [1, ), hen we have Υ(1) Υ() yielding he resul. 4 Concluding Remars Remar 4.1. By puing a = b = β = 1 ino inequaliies (14) and (16), we hus obain respecively, inequaliies (8) and (9) as in [6]. References [1] R. Díaz and E. Pariguan, On hypergeomeric funcions and Pachhammer -symbol, Divulgaciones Maemícas 15(2)(2007), [2] V. Krasniqi, T. Mansour and A. Sh. Shabani, Some Monooniciy Properies and Inequaliies for Γ and ζ Funcions, Mahemaical Communicaions 15(2)(2010), [3] V. Krasniqi and F. Merovci, Logarihmically compleely monoonic funcions involving he generalized gamma funcion, Le Maemaiche LXV(2010), [4] T. Mansour, Some inequaliies for he q-gamma Funcion, J. Ineq. Pure Appl. Mah. 9(1)(2008), Ar. 18. [5] F. Merovci, Power Produc Inequaliies for he Γ Funcion, In. Journal of Mah. Analysis 4(21)(2010), [6] K. Nanomah and M. M. Iddrisu, Some Inequaliies Involving he Raio of Gamma Funcions, In. Journal of Mah. Analysis 8(12)(2014), Received: March 19, 2014