ON CERTAIN SUBCLASSES OF UNIFORMLY SPIRALLIKE FUNCTIONS ASSOCIATED WITH STRUVE FUNCTIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The main object of this paper is to find necessary and sufficient conditions for generalized Struve functions of first kind to be in the classes S P p(α,β ) and U C S P(α,β ) of uniformly spirallike functions and also give necessary and sufficient conditions for z(2− up(z)) to be in the above classes. Furthermore, we give conditions for the integral operator L (m,c,z) = ∫ z 0 (2− up(t))dt to be in the class U C S PT (α,β ). Several corollaries and consequences of the main results are also considered.


INTRODUCTION AND DEFINITIONS
Let A denote the class of the normalized functions of the form (1) f (z) = z + ∞ ∑ n=2 a n z n , which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Further, let T be a subclass of A consisting of functions of the form, for some α with |α| < π/2 and for all z ∈ U . Also f (z) is convex spirallike if z f (z) is spirallike.
In [36], Selvaraj and Geetha introduced the following subclasses of unifromly spirallike and convex spirallike functions.
Definition 1. 1. A function f of the form (1) is said to be in the class S P p (α, β ) if it satisfies the following condition: and f ∈ U C S P(α, β ) if and only if z f (z) ∈ S P p (α, β ).
In particular, we note that S P p (α, 0) = S P p (α) and U C S P(α, 0) = U C S P(α), the classes of uniformly spirallike and uniformly convex spirallike were introduced by Ravichandran et al. [33]. For α = 0, the classes U C S P(α) and S P p (α), respectively reduces to the classes U C V and S P introduced and studied by Rønning [35].
The class R τ (A, B) was introduced earlier by Dixit and Pal [9]. If we put we obtain the class of functions f ∈ A satisfying the inequality which was studied by (among others) Padmanabhan [32] and Caplinger and Causey [6].
It is well known that the special functions (series) play an important role in geometric function theory, especially in the solution by de Branges of the famous Bieberbach conjecture.The surprising use of special functions (hypergeometric functions) has prompted renewed interest in function theory in the last few decades. There is an extensive literature dealing with geometric properties of different types of special functions, especially for the generalized, Gaussian hypergeometric functions [7,19,25,27,37,39] and the Bessel functions [1,2,3,4,26].
We recall here the Struve function of order p (see [31,40]), denoted by H p is given by , ∀z ∈ C which is the particular solution of the second order non-homogeneous differential equation where p is unrestricted real(or complex) number. The solution of the non-homogeneous differential equation is called the modified Struve function of order p and is defined by the formula , ∀z ∈ C.
Let the second order inhomogeneous linear differential equation [40], where b, p, c ∈ C which is natural generalization of Struve equation. It is of interest to note that when b = c = 1, then we get the Struve function (4) and for c = −1, b = 1 the modified Struve function (5). This permit us to study Struve and modified Struve functions. Now, denote by w p,b,c (z) the generalized Struve function of order p given by , ∀z ∈ C which is the particular solution of the differential equation (7) Although the series defined above is convergent everywhere, the function ω p,b,c is generally not univalent in U. Now,consider the function u p,b,c defined by the transformation By using well known Pochhammer symbol (or the shifted factorial) defined, in terms of the familiar Gamma function, by we can express u p,b,c (z) as . This function is analytic on C and satisfies the secondorder linear differential equation For convenience throughout in the sequel, we use the following notations and for if c < 0, m > 0 let, Now, we consider the linear operator Orhan and Yagmur [40] have determined various sufficient conditions for the parameters p, b and c such that the functions u p,b,c (z) or z → zu p,b,c (z) to be univalent, starlike, convex and close to convex in the open unit disk. Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [7,25,27,37,39]), Struve functions (see [8,23]), Poisson distribution series (see [10,14,16,28,30]) and Pascal distribution series (see [11,15,18]),, we obtain sufficient condition for function h µ (z), given by where 0 ≤ µ ≤ 1, to be in the classes S P p (α, β )and U C S P(α, β ) and also proved that those sufficient conditions are necessary for functions of the form (11). Furthermore, we give necessary and sufficient conditions for I (c, m) f to be in U C S PT (α, β ) provided that the function f is in the class R τ (A, B). Finally, we give conditions for the integral operator L (m, c, z) = z 0 (2 − u p (t))dt to be in the class U C S PT (α, β ).
To establish our main results, we need the following lemmas.
Lemma 1.2. (see [36]) (i) A sufficient condition for a function f of the form (1) to be in the class S P p (α, β ) is that and a necessary and sufficient condition for a function f of the form (2) to be in the class S P p T (α, β ) is that the condition (13) is satisfied. In particular, when β = 0, we obtain a sufficient condition for a function f of the form (1) to be in the class S P p (α) is that and a necessary and sufficient condition for a function f of the form (2) to be in the class S P p T (α) is that the condition (14) is satisfied.
(ii) A sufficient condition for a function f of the form (1) to be in the class U C S P(α, β ) is that and a necessary and sufficient condition for a function f of the form (2) to be in the class U C S PT (α, β ) is that the condition (15) is satisfied. In particular, when β = 0, we obtain a sufficient condition for a function f of the form (1) to be in the class U C S P(α) is that

MAIN RESULTS
Theorem 2.1. Let c < 0 and m > 0 . Then h µ (z) ∈ S P p (α, β ) if Proof. Since Differentiating zu p (z) with respect to z and taking z = 1 we have Also, differentiating zu p (z) + u p (z) with respect to z and taking z = 1, we have Since h µ (z) ∈ S P p (α, β ), by virtue of (13), it suffices to show that Writing n 2 = n(n − 1) + n, we get From (20), (21) and (22), we immediately have But the last expression is bounded above by cos α − β if and only if (18) holds.
Proof. By virtue of (13), it suffices to show that Since h 0 (z) = zu p (z), hence by taking µ = 0 in (23) we get the inequality (25). Hence by taking µ = 0 in the Theorem 2.1, we get the desired result given in (24). Theorem 2.3. Let c < 0 and m > 0.Then zu p (z) ∈ U C S P(α, β ) if Moreover (26) is necessary and sufficient for z(2 − u p (z)) to be in U C S PT (α, β ).
Corollary 2.5. Let c < 0 and m > 0. Then zu p (z) ∈ S P p (α) if Moreover (28) is necessary and sufficient for z(2 − u p (z)) to be in S P p T (α).
Corollary 2.6. Let c < 0 and m > 0.Then zu p (z) ∈ U C S P(α) if Moreover (29) is necessary and sufficient for z(2 − u p (z)) to be in U C S PT (α).

INCLUSION PROPERTIES
Making use of Lemma 1.3, we will study the action of the Struve function on the class U C S PT (α, β ). (A, B), and if the inequality Proof. Let f be of the form (1) belong to the class R τ (A, B). By virtue of (15), it suffices to show that Since f ∈ R τ (A, B) then by Lemma 1.3, we have Hence Making use of (21) and (22), we get but this last expression is bounded above by cos α − β if and only if (30) holds. Putting β = 0 in Theorems 3.1 and 3.2, we obtain the following corollaries.  (ii) If we put α = 0 in Corollary 2.6, we obtain the necessary and sufficient condition for z(2 − u p (z)) to be in U S T (0, 1) [23, Corollary 2, (ii)]. We note that the coefficient of u p (1) in [23, Corollary 2, (ii)], should be corrected to (3 + 2β − α).