Harmonic starlikeness and convexity of integral operators generated by Poisson distribution series

The purpose of the present paper is to establish connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the Poisson distribution series. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.


Introduction
Let A denote the class of functions f (z) of the form (1) f a n z n which are analytic in the open unit disc U = {z : z ∈ C and |z| < 1} and satisfy the normalization condition f (0) = f (0) − 1 = 0. Further, we denote by S the subclass of A consisting of functions of the form (1) which are also univalent in U .A continuous complex-valued function f = u + iv is said to be harmonic in a simply-connected domain D if both u and v are real harmonic in D. In any simply-connected domain we can write f = h + g , where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f .A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that h (z) > g (z) , z ∈ D. See Clunie and Sheil-Small [4], for more basic results on harmonic functions one may refer to the following standard introductory text book by Duren [6], (see also [1]).
Let H be the family of all harmonic functions of the form f = h + g, where Denote by S H the subclass of H functions f = h + g that are harmonic univalent and sense-preserving in the open unit disk U = {z : |z| < 1} for which f (0) = f z (0) − 1 = 0.
Note that S H reduces to class S of normalized analytic univalent functions if the co-analytic part of its member is zero.In fact, Clunie and Sheil-Small [4] investigated the class S H .We also let the subclass S 0 H and S H were first studied in [4].A function f (z) of the form (2) in S H is said to be harmonic starlike of order α, (0 ≤ α < 1) in U , if and only if and is said to be harmonic convex of order α, (0 The classes of all harmonic starlike functions of order α and harmonic convex functions of order α are denoted by S * H (α) and K H (α), respectively.These classes have been extensively studied by Jahangiri [8].
For α = 0, these classes S * H (α) and K H (α) were denoted by S * H and K H respectively.These classes were studied in detail by Silverman [16] and Silverman and Silvia [17], (see also [3]).Further, we let K 0 H , S * ,0 H and C 0 H denote the subclasses of S 0 H of harmonic functions which are, respectively, convex, starlike and close-to-convex in U .For definitions and properties of these classes, one may refer to ( [1], [4]) or [6].
Very recently, Porwal [12] introduce a power series whose coefficients are probabilities of Poisson distribution By ratio test the radius of convergence of above series is infinity.Using the above series they obtain some interesting results on certain classes of analytic univalent functions.Some other interesting results also found in [5], [9] and [10], (see also [7], [11]).The convolution (or Hadamard product) of two series f (z) = ∞ n=0 a n z n and g(z) = ∞ n=0 b n z n is defined as the power series Using the definition (5), we introduce the integral operator I : H → H by or equivalently where * denotes the usual Hadamard product or convolution of two power series.The hypergeometric series plays an important role in Geometric Function theory.Recently Ahuja [2] studied the harmonic starlikeness and convexity of integral operators generated by hypergeometric series.Analogues to these results Porwal [13],(see also [14]- [15], [18]), studied the harmonic starlikeness and convexity of integral operators generated by generalized Bessel functions.In the present paper motivated with the above mentioned work we establish a number of connections between the classes H by applying the integral operator I.

Preliminary Lemmas
To prove our main results we shall require the following lemmas.
H where h and g are given by (2) with B 1 = 0, then Lemma 2.2.( [8]) Let f = h + g be given by (2).If for some α(0 ≤ α < 1) and the inequality , where T consists of the functions f = h + g in S H so that h(z) and g(z) are of the form ( 9) Remark 2.1.In [8], it is also shown that f = h + g given by ( 9) is in the family T S * H (α), if and only if the coefficient condition (8 Lemma 2.3.( [8]) Let f = h + g be given by (2).If for some α(0 ≤ α < 1) and the inequality is satisfied, then f is harmonic, sense-preserving univalent functions in U and f ∈ K H (α).
Remark 2.2.In [8], it is also shown that f = h + g given by ( 9) is in the family T K H (α), if and only if the coefficient condition (11) holds.Moreover, H or C 0 H where h and g are given by (2) with B 1 = 0, then , n ≥ 2.

Main Results
In our first result, we determine conditions which guarantee that the integral operator I is harmonic starlike in U .
are satisfied, then the integral operator I is sense-preserving, harmonic univalent and maps H in to S * H . Proof.Note that , where H(z) and G(z) are given by (7).In order to show that I is locally univalent and sense-preserving it suffices to show that |H (z)| − |G (z)| > 0 in U .Using the condition (i), we have To show that I(f ) is univalent in U , we follow the method of Theorem 1 in [8].That is, for z 1 = z 2 in U , it suffices to prove that Since from the given condition (i), we have it follows from the given hypothesis that the last inequality is positive.Therefore, from the inequality (13) we have This proves the univalence of I(f ).
In order to prove that I(f ) ∈ S * H ≡ S * H (0), it suffices to show that P 1 ≤ 1, because of Lemma 2.2, where , because of given condition (i), we obtain This completes the proof of Theorem 3.1.
We next find a sufficient condition for which the integral operator I maps K 0 H into S * H (α). Theorem 3.2.If m j > 0 for j = 1, 2. If for some α(0 ≤ α < 1), the inequality H where h and g are given by ( 2) with B 1 = 0. We need to prove that where H and G are given by ( 7) with B 1 = 0 are analytic functions in U .In view of Lemma 2.2, it is enough to show that P 2 ≤ 1 − α, where ( 14) Applying Lemma 2.1, we have The last expression is bounded above by (1 − α) by the given hypothesis.Thus the proof of Theorem 3.2 is established.
If for some α(0 ≤ α < 1) and the inequality Proof.Let f = h + g ∈ S * ,0 H where h and g are given by ( 2) with B 1 = 0. We need to prove that where H and G are given by ( 7) with B 1 = 0 are analytic functions in U .In view of Lemma 2.2, it is enough to show that P 2 ≤ 1 − α, where Applying Lemma 2.4, we have where h and g are given by ( 9), we need to prove that the integral operator Using Remark 2.1, we obtain by the given condition and this completes the proof of the theorem.
We next explore a sufficient condition which ensures that I maps K 0 H in to K H (α). Theorem 3.5.If m j > 0, for (j = 1, 2).If for some α(0 ≤ α < 1), the inequality H where h and g are given by ( 2) with B 1 = 0. We need to prove that I(f ) = H + G ∈ K H (α) where H and G are given by ( 7) with B 1 = 0 are analytic functions in U .In view of Lemma 2.3, it is enough to show that P 4 ≤ 1 − α, where Thus the proof of Theorem 3.5 is established.
The proof of following theorems are similar to previous theorems so we state only the results.