On Certain Classes of Harmonic p-Valent Functions Defined by an Integral Operator

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 inD is that |h󸀠(z)| > |g󸀠(z)| inD (see [1]). Denote by S H the class of functions f of the form (1) that are harmonic univalent and sense preserving in the unit disc U = {z : |z| < 1} for which f(0) = f z (0) − 1 = 0. Recently, Jahangiri and Ahuja [2] defined the class


Introduction
A continuous complex-valued function = + defined in a simply connected complex domain is said to be harmonic in if both and are real harmonic in . In any simply connected domain, we can write where ℎ and are analytic in . We call ℎ the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense preserving in is that |ℎ ( )| > | ( )| in (see [1]). Denote by the class of functions of the form (1) that are harmonic univalent and sense preserving in the unit disc = { : | | < 1} for which (0) = (0) − 1 = 0.
In this paper, we obtain coefficient characterization of the classes H , ( ; ) and H − , ( ; ). We also obtain extreme points and distortion bounds for functions in the class H − , ( ; ).
Proof. According to (2) and (3), we only need to show that It follows that For = , we have Setting that the proof will be complete if we can show that | ( )| < 1.
Using the condition (7), we can write International Journal of Analysis where The harmonic functions are as follows: where ∑ ∞ = +1 | | + ∑ ∞ = | | = 1 show that the coefficient bound given by (7) This completes the proof of Theorem 1.
In the following theorem, it is shown that the condition (7) is also necessary for functions = ℎ + , where ℎ and are of the form (6).
The previous required condition (19) must hold for all values of in . Upon choosing the values of on the positive real axis where 0 ≤ = < 1, we must have If the condition (18) does not hold, then the numerator in (20) is negative for sufficiently close to 1. Hence there exists 0 = 0 in (0, 1) for which the quotient in (20) is negative. This contradicts the required condition for ∈ H − , ( ; ), and so the proof of Theorem 2 is completed.

Extreme Points and Distortion Theorem
Our next theorem is on the extreme points of convex hulls of the class H − , ( ; ) denoted by H − , ( ; ). Then, and so ∈ H − , ( ; ).

(28)
The result is sharp.
Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let The bounds given in Theorem 4 for functions = ℎ+ , whereℎ and of form (6), also hold for functions of form (2) if the coefficient condition (7) showing that the bounds given in Theorem 4 are sharp.
Remark 5. (i) Putting = 1 in the previous results, we obtain the results of Cotirla [5].