Ahmed SALEM

Department of Basic Science，F(xiàn)aculty of Information Systems&Computer Science，October 6 University，Sixth of October City，Egypt

Eid S.KAMEL

Department of Mathematics，F(xiàn)aculty Science，Al Jouf University，Sakaka，Al Jouf，Kingdom of Saudi Arabia

SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS?

Ahmed SALEM

Department of Basic Science，F(xiàn)aculty of Information Systems&Computer Science，October 6 University，Sixth of October City，Egypt

E-mail：ahmedsalem74@hotmail.com

Eid S.KAMEL

Department of Mathematics，F(xiàn)aculty Science，Al Jouf University，Sakaka，Al Jouf，Kingdom of Saudi Arabia

E-mail：Kamel-email:es kamel@yahoo.com

In this paper，the q-analogue of the Stirling formula for the q-gamma function（Moak formula）is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q＞ 0.The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number.Our results are shown to be a generalization of results which were obtained by Selvi and Batir［23］.

completely monotonic functions；inequalities；q-gamma function；q-polygamma function

2010 MR Subject Classification 33D05；26D15；26A48

1 Introduction

In the recent past a lot of papers appeared providing inequalities and complete monotonicity properties for the gamma function，q-gamma function and related functions；see［2-4，6-10，14，15，17，20-23］and the references given therein.Sevli and Batir［23］concerned the function

and some other functions related toμ（x），whereμ（x）is the classical remainder in the Stirling formula for gamma function given as

It is well known thatμ（x）is completely monotonic on（0，∞）.They obtained some complete monotonicity results and as applications of these they offer upper and lower bounds for thegamma function and harmonic numbers.Many of the classical facts about the ordinary gamma function were extended to the q-gamma function（see［5，12，13，16，19］and the references given therein）.The q-gamma function is defined for positive real numbers x and q/=1 as

and

From the definitions，for a positive x and q≥1，we get

An important fact for gamma function in applied mathematics as well as in probability is the Stirling formula that gives a pretty accurate idea about the size of gamma function.With the Euler-Maclaurin formula，Moak［12］obtained the following q-analogue of Stirling formula（see also［17］）

where H（·）denotes the Heaviside step function，Bkare the Bernoulli numbers，?q=q if 0＜q≤1 and?q=q-1if q≥1［x］q=（1-qx）/（1-q），Li2（z）is the dilogarithm function defined for complex argument z as［1］

Pkis a polynomial of degree k satisfying

and

where r=exp（4π2/logq）.It is easy to see that

and so（1.6）when letting q→1，tends to the ordinary Stirling formula［1］

An important related function to q-gamma function is the q-digamma function（q-Psi function ψq）defined as the logarithmic derivative of the q-gamma function

From（1.3），we get for all real variable x＞0，

Krattenthaler and Srivastava［11］proved that ψq（x）tends to ψ（x）when letting q→1 where ψ（x）is the ordinary Psi（digamma）function.For more details on q-digamma function see［18］.

A real-valued function f，defined on an interval I，is called completely monotonic，if f has derivatives of all orders and satisfies

These functions have numerous applications in various branches，like，for instance，numerical analysis and probability theory.From（1.13）and（1.14），Alzer and Grinshpan［3］concluded that ψ′q（x）is strictly completely monotonic on（0，∞）for any q＞0，that is，

Selvi and Batir［20］proved the complete monotonicity properties of some functions involving the gamma and polygamma functions.As consequences of them they established various inequalities for the gamma function and the harmonic numbers.The main purpose of this paper is to generalizing of their results for q＞0 after replacing the ordinary concepts by its q-analogues in q-calculus.Further results are also derived.

2 The Main Results

According to Moak formula for q-gamma function（1.6），the q-analogue ofμ（x）defined in（1.2）can be denoted and defined as

It is useful to prove thatμq（x）is completely monotonic on（0，∞）for q＞0；this follows from the theorem.

Theorem 2.1 The functionμq（x）as defined in（2.1）is completely monotonic on（0，∞）for all real q＞0.

Proof Differentiation yields

When 0＜q＜1，（1.13）gives

where

From the previous relations，we can conclude that

which reveals that-μ′q（x）is completely monotonic on（0，∞）. When q≥1，（1.5）and（2.2）give

As a consequence of the previous theorem and the fact that

we obtain the following bounds for the q-gamma function.

Corollary 2.2 Let x and q be positive real numbers.Then we have

and

where

A q-analogue of Harmonic number defined by［24］as

which can be related to ψq（n+1）for a positive integer n by

we can deduce the following.

Corollary 2.3 Let x and q be positive real numbers.Then we have

and for a positive integer n，we have

Lemma 2.4 The function

Proof Differentiaition gives y log2yg′（y）=h（y）where

Differentiation again gives y logy（2（1-y）+logy+y logy）2h′（y）=f（y），where

Using the same technique used to prove Lemma 1.1 in［17］，we can write f（y）as

where

Theorem 2.5 Let x，q＞0 and a≥0.The function

is completely monotonic if and only if a≥g（?q）where g（q）as defined in（2.11）.Also，the function-Fb（x；q）is completely monotonic if and only if b=0.

Proof Differentiation yields for q＞0

When 0＜q＜1 and x＞0，we conclude that

where

It is obvious that the function a→f（a，y），0＜y＜1 is decreasing and it has just one zero function depending on y at a=g（y），0＜y＜1 where the function g defined as in（2.11）.The function f（0，y）can be computed after short calculations as

When q≥1 and x＞0，（1.5）and（2.14）give

These conclude that-F′a（x；q）；a≥g（?q）and F′0（x；q）are completely monotonic on（0，∞）for q＞0.This means that Fa（x；q）；a＞g（?q）is decreasing on（0，∞）for q＞0 and F0（x；q）is increasing on（0，∞）for q＞0.From（1.13）and（1.5）we get

Conversely，let Fa（x；q）is completely monotonic on（0，∞）for q＞0，then?q-xFa（x；q）≥0. Salem［17］proved that

and from（1.13）we get

These conclude that

or equivalently

From the above theorem，we can obtain F0（x；q）＜0＜Fg（?q）（x；q），x＞0 from which we provide the following.

Corollary 2.6 Let x and q be positive real numbers.Then we have

with the best possible constants α=g（?q）and β=0 for q＞0.

Theorem 2.7 Let x and q be positive real numbers.Then the function

is completely monotonic on（0，∞）for q＞0.

Proof Differentiation yields

When 0＜q＜1 and x＞0 we conclude that

where

which can be read as

where

and

Let?is the forward shift operator and?i=?（?i-1），i=1，2，···.It is easy to see that?3?2（n）=3?32?n-6＞0 for all n≥4 and?2?2（4），??2（4）and ?2（4）are greater than zero which lead to ?2（n）＞0 for all n≥4.These yield that the function ?（y）＞0；y∈（0，1）. Therefore the function-G′q（x）is completely monotonic on（0，∞）.

When q≥1 and x＞0，（1.5）and（2.17）give

which leads to-G′q（x）is completely monotonic on（0，∞）.These conclude that-G′q（x）is completely monotonic on（0，∞）for q＞0.This means that Gq（x）is decreasing on（0，∞）.To complete the proof，it suffices to prove thatTo do this，we havefor q＞0 and when 0＜q＜1，we get

and when q＞1，we get

These end the proof.

Corollary 2.8 Let x and q be positive real numbers.Then we have

Corollary 2.9 Let x and q be positive real numbers.Then we have

and for a positive integer n，we have

where

The proofs of the previous two corollaries come from the monotonicity of Gq（x）and G′q（x）and the facts

and

Theorem 2.10 Let x and q be positive real numbers.Then the function

is strictly completely monotonic on（0，∞）for q＞0.

Proof Differentiation yields

When 0＜q＜1 and x＞0 we conclude that

where

which can be rewrite，after short calculations，as

This means that K′q（x）＜0；x∈（0，∞）.Thus it follows from（2.23）that-K′q（x）is strictly completely monotonic on（0，∞）for 0＜q＜1.

to compute that

Using the previous result to calculate the limit

Corollary 2.11 Let x and q be positive real numbers.Then we have

Corollary 2.12 Let x and q be positive real numbers.Then we have

and

Similarly，the proofs of the previous two corollaries come from the monotonicity of Kq（x）and K′q（x）and the facts

［1］Abramowitz M，Stegun C A.Handbook of Mathematical Functions with Formulas，Graphs，Mathematical Tables.Applied Mathematics Series，Vol 55.Washington，DC:National Bureau of Standards，1964

［2］Alzer H，F(xiàn)elder G.A Tur′an-type inequality for the gamma function.J Math Anal Appl，2009，350:276-282

［3］Alzer H，Grinshpan A Z.Inequalities for the gamma and q-gamma functions.J Appr Theory，2007，144: 67-83

［4］Alzer H，Batir N.Monotonicity properties of the gamma function.Appl Math Lett，2007，20:778-781

［5］Askey R.The q-gamma and q-beta functions.Appl Anal，1978，8:125-141

［6］Elezovic N，Giordano C，Pecaric J.Convexity and q-gamma function.Rendiconti del Circolo Matematico di Palermo，Series II，1999，48:285-298

［7］Gao P.Some monotonicity properties of gamma and q-gamma functions.ISRN Math Anal，2011，2011: 1-15

［8］Grinshpan A Z，Ismail M E H.Completely monotonic functions involving the gamma and q-gamma functions.Proc Amer Math Soc，2006，134:1153-1160

［9］Ismail M E H，Muldoon M E.Inequalities and monotonicity properties for gamma and q-gamma functions//Zahar R V M，ed.Approximation and Computation，International Series of Numerical Mathematics. Boston，MA:Birkhauser，1994，119:309-323

［10］Ismail M E H，Lorch L，Muldoon M E.Completely monotonic functions associated with the gamma function and its q-analogues.J Math Anal Appl，1986，116:1-9

［11］Krattenthaler C，Srivastava H M.Summations for basic hypergeometric series involving a q-analogue of the digamma function.Comput Math Appl，1996，32（2）:73-91

［12］Moak D S.The q-analogue of Stirling's formula.Rocky Mountain J Math，1984，14:403-413

［13］Moak D S.The q-gamma function for q＞1.Aequationes Math，1980，20:278-285

［14］Mortici C.Improved asymptotic formulas for the gamma function.Comput Math Appl，2011，61:3364-3369

［15］Mortici C.Estimating gamma function by digamma function.Comput Math Appl，2010，52:942-946

［16］Olde Daalhuis A B.asymptotic expansions of q-gamma，q-exponential and q-Bessel functions.J Math Anal Appl，1994，186:896-913

［17］Salem A.A completely monotonic function involving q-gamma and q-digamma functions.J Appr Theory，2012，164:971-980

［18］Salem A.Some properties and expansions associated with q-digamma function.Quaest Math，2013，36（1）: 67-77

［19］Salem A.A q-analogue of the exponential integral.Afrika Matematika，2013，24（2）:117-125

［20］Salem A.An infinite class of completely monotonic functions involving the q-gamma function.J Math Anal Appl，2013，406（2）:392-399

［21］Salem A.Two classes of bounds for the q-gamma and the q-digamma functions in terms of the q-zeta functions.Banach J Math Anal，2014，8（1）:109-117

［22］Salem A.Complete monotonicity properties of functions involving q-gamma and q-digamma functions. Math Inequalities Appl，2014，17（3）:801-811

［23］Sevli H，Batir N.Complete monotonicity results for some functions involving the gamma and polygamma functions.Math Comput Modelling，2011，53:1771-1775

［24］Wei C，Gu Q.q-generalizations of a family of harmonic number identities.Adv Appl Math，2010，45:24-27

?Received April 24，2013.