ON TUR ´ A N’S TYPE INEQUALITIES FOR SOME SPECIAL FUNCTIONS AND OPERATORS

The purpose of this paper is to use the generalized Schwarz inequality to give a short proofs of new Tur ´ a n-type inequalities for some special functions and operators.

The proof given by Turán is based on the recurrence relation [5, p. 81], and on the differential relation [5, p. 83] [4] derived Turán-type inequalities for the positive zeros C νK , K = 1, 2, ... of the general Bessel function where J ν (x) and Y ν (x) denote the Bessel function of the first and the second kind respectively.Finally, the corresponding results for the positive zeros ] and for the zeros of Hermite Laguerre and ultraspherical polynomials have been established in [1] [2] and [3] respectively.The object of this paper is to prove new Turán-type inequalities for some special functions like unified Riemann zeta function, Negapolygamma function, generalized Hurwitz-Lerch Zeta function and some operator like Mellin transform, Riemann-Liouville fractional integral operator.
The approach used in this paper is based, prevalently on Sturm theory and is different from the one used in the above mentioned papers.Here our main tool is the following generalized Schwarz inequality.
where f and g are two non negative functions of a real variable and m and n belonging to a set R of real numbers, such that the integral in (1.5) exists.

Main Results
Theorem 2.1.The Legendre polynomial of order n is denoted by P n (x) in [10].Then Proof.From [13, pp.203], we have and (1.5) with n − 1, n + 1 instead of m, n, to establish the inequality for x ≥ 1, and then use the fact that P n (x) is ever for n even, odd for n odd, to prove it for x ≤ −1.
This completes the proof. 2 Theorem 2.2.The unified Riemann zeta function is denoted by φ µ (z, s, a) in [7, p. 87].Then Proof.For s > 1, the unified Riemann zeta function satisfies the integral relation We obtain (2.5) Furthermore using (2.4) the inequality becomes or by the functional relation Γ(s + 1) = sΓ(s), This completes the proof. 2 Remark 2.1.In the above theorem(2.2),if we put a = µ = z = 1 then the given relation satisfies for the Riemann zeta function.

Proof.
For z > 0, the negapolygamma function can be written in integral form as when n is an arbitrary positive integer.We write the above integral in suitable form and similar steps used like Theorem(2.2)andusing (1.5).We arrive at the required result. 2 Theorem 2.4.The Mellin transform of a function f (x) is defined as where s+p 2 is an integer.
Proof.In the theorem, we choose the integer p and s both even or odd in such a way that p+s 2 is an integer.By (1.5), we get This completes the proof. 2 Theorem 2.5.The Riemann -Liouville fractional integral operator [8] of a function f (x) is defined as

Proof.
We use the similar lines as used in the above theorems and get the proof of this theorem.2 The generalized Hurwitz-Lerch Zeta function φ α,β;γ (z, s, a) is recently defined by Garg, Jain and Kalla in [12] as follows: Proof.
We use the similar lines as used in the above theorems and get the proof of this theorem.2