On some properties of generalized Struve function

The main purpose of this investigation is to present some monotonic and log-concavity properties of the generalized Struve function. By using Hadamard product representation of the generalized Struve function, we investigate the sign of this function on some sets. Also, we determine an interval such that the generalized Struve function is decreasing in this interval. Moreover, we show that generalized Struve function is strictly logaritmically concave on some intervals. In addition, we prove that a function related to generalized Struve function is increasing function on R.


Introduction and Preliminaries
In the last three decades many geometric and monotonic properties of some special functions like Bessel, Struve, Lommel, Mittag-Le-er, Wright functions and their generalizations were investigated by many authors. In general, by using the properties of zeros of the special functions many mathematicians studied about univalence, starlikeness, convexity and close-to-convexity of the mentioned functions. In addition, some authors focused on the monotonicity and log-convexity properties of the special functions by using their integral representations and some earlier results on analytic functions. For more information about these investigations the readers are referred to the papers [1,2,3,4,5,8,9,10,11] and references therein. Some inequalities which were obtained via above special functions and monotonic properties of these functions are intensively used in engineering sciences, mathematical physics, probability and statistics, and economics. Especially, it is known that the logarithmic concavity and logarithmic convexity properties have an important role in economics. Information on the logarithmic concavity and logarithmic convexity in the economic can be found in [7] and its references, comprehensively. In this study, motivated by the some earlier studies, our main goal is to give some monotonic and log-concavity properties of the generalized Struve functions. 348 · IBRAH · IM AKTAŞ, HAL · IT ORHAN, AND DORINA R µ ADUCANU It is well-known that many special functions can be de…ned by using familiar gamma function. That is why, we want to remember the de…nition of gamma function. The Euler's gamma or classical gamma function is de…ned by the following improper integral, for x > 0: On the other hand, the de…nition of logarithmic concavity of a function can be given as follow: . A function f is said to be log-concave on interval (a; b) if the function log f is a concave function on (a; b).
The log-concavity of the function f on the interval (a; b) can be shown by using one of the following two conditions: ii. log f 00 < 0: Also the following lemma due to Biernacki and Krzy· z (see [6]) will be used in order to prove some monotonic properties of the mentioned functions.

Lemma 2.
Consider the power series f (x) = P n 0 a n x n and g(x) = P n 0 b n x n , where a n 2 R and b n > 0 for all n 2 f0; 1; : : : g, and suppose that both converge on ( r; r); r > 0. If the sequence f an bn g n 0 is increasing(decreasing), then the function is also increasing(decreasing) on (0; r).
It is important to note that the above result remains true for the even or odd functions.

Main Results
In this section, we are going to discuss some properties like monotonicity and log-concavity of the generalized Struve function by using its product representation which is known as Hadamard product or Weierstrassian decomposition. The generalized Struve function has the following series representation (see [12]): for q 2 N; p; b; c 2 C and > 0. The author studied some geometric properties such as starlikeness and convexity of generalized Struve function in [12]. Also, the author showed that the zeros of the generalized Struve function are all real. In the same paper, by using Hadamard's theorem an in…nite product representation of the generalized Struve function was given as follow (see [12, Lemma 2.1]):   Proof. By using the fact that the product of two strictly log-concave function is also strictly log-concave, it is possible to prove the log-concavity of the generalized Struve function S q p;b;c; (x) on 3 : Because of this, we consider the function S q p;b;c; (x) in the following form: We known from part c. of Theorem 3 that the generalized Struve function q p;b;c; (x) is strictly log-concave on 3 : In addition, since As a result, the function S q p;b;c; (x) is strictly log-concave on (0; q x p;b;c; ;1 ) [ 2 as a product of two strictly log-concave functions. Now, let de…ne the function x 7 ! h q p;b; (x) by putting c = 1 in (3). It is easily seen that the function h q p;b; (x) has the following in…nite sum representation: By using the Lemma 2 we have the following: The function is increasing on (0; 1) for p; b; ; q 2 R + .
Proof. By using the in…nite sum representation of the function h q p;b; (x) which is given by (5), it can be written that ; where A n = 2n ( p + b+2 2 ) n!4 n (nq + p + b+2 2 ) and B n = ( p + b+2 2 ) n!4 n (nq + p + b+2 2 ) : 352 · IBRAH · IM AKTAŞ, HAL · IT ORHAN, AND DORINA R µ ADUCANU Cauchy-Hadamard theorem for power series implies that the both series P 1 n=0 A n x 2n and P 1 n=0 B n x 2n are convergent for all x 2 R; since lim n!1 Moreover, we can say that A n 2 R and B n > 0 for all n = 0; 1; 2; : : : : On the other hand, if we consider the sequence H n = A n B n = 2n; then we deduce H n+1 H n = n + 1 n > 1: So the sequence fH n g n 0 is increasing. As a result, by applying the Lemma 2 to the function x 7 !
x(h q p;b; (x)) 0 h q p;b; (x) the proof is completed.