Certain subclasses of starlike functions of complex order involving the Hurwitz – Lerch Zeta function

Making use of the Hurwitz–Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by TS b (α, β, γ) and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class TS b (α, β, γ). In particular, we obtain integral means inequalities for the function f(z) belongs to the class TS b (α, β, γ) in the unit disc.

where T is given by (1.2) and Lf (z) is given by Lf (z where T is given by (1.2) and The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness and convexity for the class T S μ b (α, β, γ).Further, we obtain neighbourhood results and integral means inequalities for aforementioned class.

Basic properties.
In this section we obtain a necessary and sufficient condition for functions f (z) in the class T S μ b (α, β, γ).

Theorem 2.1. A necessary and sufficient condition for
where If we let z → 1 along the real axis, we have The simple computational leads the desired inequality Conversely, suppose that (2.1) is true for z ∈ U, then which completes the proof.
In the following theorem we give extreme points for the functions of the class T S μ b (α, β, γ).Theorem 2.3 (Extreme points).Let , where λ n ≥ 0 and ∞ n=1 λ n = 1.The proof of the Theorem 2.3 follows the lines similar to the proof of the theorem on extreme points given by Silverman [25].
Proof.Given f ∈ T, and f close-to-convex of order δ, we have For the left hand side of (3.1) we have The last expression is less than Using the fact that f ∈ T S μ b (α, β, γ) if and only if we can say (3.1) is true if > δ, where > δ, where Each of these results are sharp for the extremal function f (z) given by (2.3).
Proof.Given f ∈ T such that f is starlike of order δ, we have For the left hand side of (3.2) we have Using the fact that f ∈ T S μ b (α, β, γ) if and only if Or, equivalently, which yields the starlikeness of the family.Using the fact that f is convex if and only if zf is starlike, we can prove (2), on lines similar to the proof of (1).[26], the following subordination result will be required in our present investigation.Lemma 4.1 ([18]).If the functions f (z) and g(z) are analytic in U with g(z) ≺ f (z), then

Integral means. Motivated by Silverman
n=2 is a non-decreasing sequence, then for z = re iθ and 0 < r < 1, we have and Proof.Let f (z) be of the form (1.2) and By Lemma 4.1, it suffices to show that From (4.3) and (2.1), we obtain This completes the proof of Theorem 4.2.

Inclusion relations involving N δ (e).
To study the inclusion relations involving N δ (e) we need the following definitions.Following [2,8,19,24], we define the n, δ neighbourhood of the function f (z) ∈ T by Particulary for the identity function e(z) = z, we have On the other hand, from (2.1) and (5.4) we have (5.5) then N δ (g) ⊂ T S μ b (α, β, γ, η).
Proof.Suppose that f ∈ N δ (g), then we find from (5.1) that which implies that the coefficient inequality Next, since g ∈ T S μ b (α, β, γ), we have So that provided that η is given by (5.7).Thus by definition, f ∈ T S μ b (α, β, γ, η) for η given by (5.7), which completes the proof.
Concluding remarks.By suitably specializing the various parameters involved in Theorem 2.1 to Theorem 5.2, we can state the corresponding results for the new subclasses defined in Example 1 to Example 4 and also for many relatively more familiar function classes.
A denote the class of functions of the form (1.1) f (z) = z + ∞ n=2 a n z n which are analytic and univalent in the open disc U = {z : z ∈ C, |z| < 1}.Also denote by T a subclass of A consisting of functions of the form (1.2)