ON SOME SUBCLASSES OF STRONGLY STARLIKE ANALYTIC FUNCTIONS

. The aim of the present article is to investigate a family of univalent analytic functions on the unit disc D deﬁned for M ≥ 1 by Some proprieties, radius of convexity and coeﬃcient bounds are obtained for classes in this family.


Introduction
Let A be the set of analytic function on the unit disc D with the normalization f (0) = f (0) − 1 = 0.
f ∈ A if f is of the form (1. 1) f (z) = z + +∞ n=2 a n z n , z ∈ D.
S denotes the subclass of A of univalent functions. A function f ∈ S is said to be strongly starlike of order α, 0 < α ≤ 1, if it satisfies the condition This class is denoted by SS * (α) and was first introduced by D. A. Brannan and W. E. Kirwan [1] and independently by J. Stankiewicz [9].
SS * (1) is the well known class S * of starlike functions. Recall that a function f ∈ S belongs to S * if the image of D under f is a starlike set with respect to the origin or, equivalently, if A function f ∈ S belongs to SS * (α) if the image of D under zf (z) f (z) lies in the angular sector Ω α = z ∈ C, Argz < απ 2 .
Let B denotes the set of Schwarz functions, i.e. ω ∈ B if and only ω is analytic in D, ω(0) = 0 and ω(z) < 1 for z ∈ D. Given two functions f and g analytic in D, we say that f is subordinate to g and we .
We obtain from the Schwarz lemma that if f ≺ g then f (0) ≤ g (0) . As a consequence of this statement, we have where a 2 and b 2 are respectively the second coefficients of f and g.
W. Janowski [2] investigated the subclass J. Sókol and J. Stankiewicz [8] introduced a subclass of SS * ( 1 2 ), namely, the class S * L defined by L 1 is the interior of the right half of the Bernoulli's lemniscate w 2 − 1 = 1.
In the present paper we are interested to the family of subclass of S is the interior of the right half of the Cassini's oval w 2 − M = M . For the particular case M = 1, S * L (1) stands for the class S * L introduced by J. Sókol and J. Stankiewicz [8]. Since L M ⊂ Ω( 1 2 ), all functions in S * L (M ) are strongly starlike of order 1 2 .
Note that all classes above correspond to particular cases of the classes of S * (ϕ) introduced by W. Ma and D. Minda [3], where ϕ is Analytic univalent function with real positive part in the unit disc D, ϕ D is symmetric with respect to the real axis and starlike with respect to ϕ(0) = 1 and ϕ (0) > 0.
Let m = 1 − 1 M and ϕ m be the function where the branch of the square root is chosen so that ϕ m (0) = 1. We have Observe that S * L corresponds to m = 0 so that S * L = S * ( √ 1 + z).
2. Some properties of the class S * L (M ) Let P the class of analytic functions p in D with p(0) = 1 and p(z) > 0 in D. For M ≥ 1, let It is easy to see that Proof. (2.1) is an immediate consequence of the Remark 2.1 Let f m ∈ A be the unique function such that Evaluating the integral in (2.3), we get f 0 is extremal function for problems in the class S * L (see [8]). It is easy to see that We need the following result by St. Ruscheweyh [5] Lemma 2.1.
Corollary 2.1. Let f belongs to S * L (M ) and |z| = r < 1, then Proof. We have We need the two following lemmas by Janowski [2]: , Theorem 1] For every P (z) ∈ P(M ) and |z| = r, 0 < r < 1, we have The infimum is attained by   Proof. Let f ∈ S * L (M ). From (3.2), there exists P ∈ P(M ) such that (3.8) can be written This yields for |z| = r, 0 < r <, Replacing (3.3) and(3.5 in (3.9), we obtain To finish, we observe that the equation h M (r) = 0 is equivalent in the interval (0, 1) to the equation Substituting in (3.7), we obtain Solving this equation in (0, 1), we get We have

Coefficient bounds for
Proof. If f ∈ S * L (M ) there exists ω ∈ B such that For 0 < r < 1 we have Replacing (4.3) in the right side of (4.5) we obtain (1 − m)n 2 |a n | 2 r 2 .
If we let r → 1 − , we obtain from le last inequality (1 − m)n 2 |a n | 2 which gives, Since ( The following corollary is an immediate consequence of (4.2).
This estimations are sharp.
Proof. If f ∈ S * L (M ) there exists ω(z) = ∞ n=1 C n z n ∈ B such that Equating coefficients for n = 2, n = 3 in both sides of (4.9), we obtain
which give respectively the sharpness of estimations (ii) and (iii).
Remark 4.2. The estimation (i) can be obtained directly from (2.6).
Remark 4.3. If we take m = 0 in Theorem 4.2, we obtain as particular case Theorem 2 [6].

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.