Some New Inequalities for the Generalized 𝜖 – Gamma, Beta and Zeta Functions

Some New Inequalities for the Generalized – Gamma, Beta and Zeta Functions

R. Suryanarayana And Ch.Gopala Rao

Dept. of Mathematics, GMR Institute of Technology,Rajam-532127, Srikakulam, A.P, India

Abstract

In this paper, we establish some properties and inequalities for the – generalized functions which are – Gamma function, Beta function and Zeta function and has

, = , Re(x )> 0,Re (y) > 0 (1.5)

+

And -Zeta function as

=0 +

, = 1 , , > 0, s > 1 (1.6) The function , satisfies the equality

given some identities which they satisfy. This inequality

leads to new inequalities involving the Beta, Gamma and Zeta functions and a large family of functions. The gamma

, = 1 1 1

0

which follows

1 1

(1.7)

and Beta functions belong to the category of the special

, = 1 , . (1.8)

transcendental functions and we will see that some mathematical constants are occurring in its study.

We mention that lim , (, ) and – Zeta

function is a generalization of Hurwitz Zeta function

, = 1 which is a generalization of the

Keywords: – generalized Gamma function , Beta function and – Zeta function.

1. Introduction

   The generalized – Gamma function as

   =0 +

   Reimann Zeta function

   =1

   = 1 . The motivation to study properties of

   generalized Gamma and – Beta functions is the fact

   =lim

   ! 1

   ,

   ,k>0,x

   (1.1)

   that ,

   appears in the combinatorics of creation and

   where ,

   is the – Pochammer symbol and is given by

   annihilation operators[3]. Recently M. Mansour [4]

   ,

   (1.2)

   =x(x+ )(x+2 )…(x+(n-1) ),x , ,n +

   determined the generalized Gamma function by a combination of some functional equations.

   In this paper , we use the definitions of the above

   It is obvious that 1, where is known as Gamma function. Also for Re (x) > 0, it holds

   0

   = 1 (1.3)

   And it follows that

   1

   generalized functions to prove a formula for 2 which is a generalization of the Legendre duplication formula for and to prove inequalities for the function , , for x,y, > 0 and x + y and the

   product 1 , for 0 < x, < 1.we also give

   = . (1.4)

   In this paper [1],[2],[3] introduced the – Beta function

   , as

   monotonicity properties for = , where

   , = and , for s 2.

   We mention that using (1.4) the following inequalities hold:

   = 1 ,>0,a (1.9)

   = 1( 1)!, >0,n (1.10) = 1, >0, (1.11)

   2 1 2 !

   holds. It is known [1] that () is completely monotonic for x > 0, so from (2.4) it follows the desired result.

   Remark 2.1. (i) From (2.3) it follows that is

   logarithmic convex on (0, ) which is proved in [2], (ii) Theorem 2.1 is a generalization of the known [1] result

   ( 2 + 1 2)= 2 2 ! , >0,n (1.12)

   also , using (1.5) and (1.8) the following equalities hold:

   +

   + , = , , , +

   that the function () is completely monotonic.

   Result 2.1. For x > 0 the function , = log

   satisfies the differential equation

   =

   +

   , x,y, > 0. (1.13)

   22

   , + 2

   2 , +

   3 , =

   , = 1 and , = 1, x ,y, > 0. (1.14)

   (2.5)

   , = 1 , , a,b, > 0 (1.15)

   Proof: From (2.1) taking the first and second derivative of

   , with respect to , we obtain

   , = 1 1 !2 , > 0, n (1.16)

   1 1

   21 !

   , =

   2 + 2 / (2.6)

   2 , = 2 3 + 1 + / +

2. The Function

   3

   3

   2

   2

   (/) (2.7)

   Theorem2.1: let x, > 0 and be the logarithmic derivative of . Then the function () is completely monotonic.

   Proof: From (1.4) , we get

   3

   From (2.3), (2.6) and (2.7), we get (2.5)

   Theorem2.2: The function satisfies the equality

   2 1

   2 = 2 + /2 (2.8)

   Log = 1 + log / or by setting

   , = Log , we obtain

   For x with Re(x) > 0.

   Proof: From (1.7) it follows that

   , = 1 + log / (2.1)

   1 1 1

   1

   , = 0

   1

   We get , , = 1 + / (2.2)

   1+

   2 1 2

   1

   Or by setting t =

   2 , , =

   22 1

   0 1

   We remind that / = . from (2.2)

   taking the derivative with respect to x, we have

   or by setting 2 = ,

   1 1 11

   1

   we obtain , =

   22 1

   0 2

   1

   =

   2 = 1 (/) (2.3)

   3 , = 1 (/)

   2

   1

   22 1

   , =

   , Or

   1

   22 1 2

   , = 1

   2

   By induction, we obtain +1 , = 1 (/)

   22 1

   (2.9)

   +

   2

   Or if we call = , , then the equation

   from (1.9) for a=1/2 , we get = , since 1 =

   = 1 (/) (2.4)

   2 2

   , from (2.9) and (1.5) , we get the equality (2.8).

   Remark 2.3. Theorem 2.2 is a generalization of the legendre duplication formula of (x).

   Repeating the same , we get (3.4) ,

   since s (s+1) .(s+n-1) = ,1.

   In[2]it was proved that

3. The Function ,

   Theorem3.1 (i) Let x, > 0 and s > 1. Then the positive

   2 , = 1

   =0 + 2

   From (1.6) for s + 2 and (3.7), we get

   (3.7)

   function x, s decreases with respect to x and also

   2

   , = , 2 (3.8)

   decreases with respect to . (ii) let x > 0 and s > 1. Then the positive function x, s decreases with respect to s for x > 1 and > 0, 0 and increases with respect to s for > 0, 0 < < 1/ and 0 < x < 1 – .

   Proof: From (1.6) we obtain

   Differentiating (3.7) with respect to x and using (3.1) for s = 2, we get

   3 , = 1 2 2 , 3

   and 4 , = 1 23! , 4

   By induction, we obtain (3.5). The equation (3.6) follows

   , =

   1 , , > 0, s >1

   from the definition(1.6), since

   =0 + +1

   Or

   , = 1

   + 1

   = 1 + + , .

   , = – , + 1 (3.1)

   + +

   =0

   , =

   = -s

   , > 0,

   4. Inequalities For , ,

   =0 + +1

   =1 + +1

   s >1 (3.2)

   Then (3.1) and (3.2) , prove the theorem 3.1(i)also the

   Theorem 4.1: le x,y, > 0 and x + y . Then the function B x, y satisfies the inequalitie

   definition(1.6) gives

   +

   2

   2

   +

   < B x, y < 122

   (4.1)

   , =

   ln (+ )

   (3.3)

   +

   +

   =0 +

   Lemma 4.1: The function B(x,y) satisfies the

   If x >1 then x > 1- , for , > 0 thus ln(x + ) > 0 so

   from (3.3) it follows that the function , decreases

   inequalities

   22( + ) < B(x, y) < 122( + ) , x,y > 0, x+y 1 (4.2)

   with s > 1 and if 0 < < 1/ and 0 < x < 1- then ln(x

   + 1

   +1

   + ) < 0 from (3.3) it follows that the function ,

   increases with s > 1.

   Result 3.1: Let x > 0, > 0 and s > 1. Then the function

   , satisfies the identities:

   Proof: The function B(x,y) is defined [1] by the integral B(x,y) = 1 1 1 1

   0

   Which can be written as

   , = 1

   , + (3.4)

   B(x,y)= 1/2 1 1 1 + 1

   1 1 1

   ,1

   0 1/2

   , = 1 , , 2 (3.5)

   (4.3)

   1 !

   And + , = , 1/ (3.6)

   If 0 < t < ½ then t < 1- t , so that the following inequalities

   1 1

   Proof: From (3.1) we obtain

   2 , = , + 1 = 1 2 + 1 , + 2

   hold 2 +2 < 2 1 1 1 <

   0

   0 0

   1/2 1 +2 (4.4)

   1/2

   and if ½ < t < 1 then 1 t < t, so that the following inequalities hold

   (4.8)

   Proof: By setting y = + 1 x instead of y in (4.1) we

   1

   1/2

   1 +2 < 1

   1

   1 1 1 <

   +2

   obtain

   211/ < B x, + 1 < 1 211/ (4.9)

   1/2

   (4.5)

   Using (1.5) the inequalities (4.9) become

   From (4.3), using the inequalities (4.4) and (4.5) and evaluating the integrals on the left and right side of the above inequalities , we obtain the inequalities (4.2) .

   Proof of theorem 4.1:By setting x / and y/ , instead of x

   211/ < +1 < 1 211/ (4.10)

   +1

   From (1.4) we obtain

   + 1 = (1 ) 1

   11

   and + 1 = 1 = 1/ .

   and y respectively in (4.2) and taking in account the relation (1.8) we get the inequalities (4.1).

   Corollary 4.1: Let x,y, > 0. Then the function B x, y

   satisfies the inequalities:

   From (4.10) using the above equalities we obtain the inequalities (4.8).

   References

   +

   21

   +

   < B x, y < 121

   (4.6)

   1. M.Abramowitz, I.A. Stegun (Eds) Hand book of

      1 +

      Or 2

      1 +

      < B x, y < 12

      (4.7)

      mathematical functions with formulae and mathematical tables, Applied Math. Series, Vol.55,4th edition with

      Proof: The above inequalities follow from (4.1) by setting x + (or y + ) instead of x (or y) and taking in account relations (1.13).

      Corollary 4.2: Let 0 < x < 1 and 0 < < 1. Then the

      following inequalities for the product 1

      corrections. Nat. Ber. Of Standards, Washington(1965).

   2. R.Diaz and E.Pariguan , On hypergeometric functions and k- Pochhammer symbol, arXiv:math. CA/0405596v2(2005).

   3. R.Diaz and E.Pariguan, Quantum symmetric functions

      holds

      2 11/ 1/ 2

      < 1 <

      1

      11/ 1

      1/ 2 1

      ,arXiv : math. QA/0312494(2003)

   4. M. Mansour, Determining the k-generalized Gamma

      function by functional equations , Int. J. Contemp. Math. Sciences, 4, (2009), 1037-1042.

      1

      1

   5. M. Abramowitz and I. tegun, Handbook of Mathematical Functions, Dover, New York, (1964) .