On subordination for classes of non-Bazilevič type

We give some subordination results for new classes of normalized analytic functions containing differential operator of non-Bazilevič type in the open unit disk. By using Jack’s lemma, sufficient conditions for this type of operator are also discussed.

(1.1) From (1.1), assuming α to be a parameter with the values α := n+m m , m ∈ N, and having n = 0 in the first term of the series, we can write F in the form (1.2) By employing (1.2), we define classes of analytic functions of fractional power.
Let A + α be the class of all normalized analytic functions F in the open disk U of the form satisfying F (0) = 0 and F (0) = 1.Moreover, let A − α be the class of all normalized analytic functions F in the open disk U of the form a n,α z n+α−1 , a n,α ≥ 0; n = 2, 3, . . ., satisfying F (0) = 0 and F (0) = 1.Definition 1.1 (Subordination Principle).For two functions f and g analytic in U, we say that the function f is subordinate to g in U and write f (z) ≺ g(z) (z ∈ U ), if there exists a Schwarz function w(z) analytic in U with w(0) = 0, and |w(z Now we define a differential operator as follows: (1.3) Let A be the class of analytic functions of the form f (z) = z + a 2 z 2 + . . . .Obradovič [8] introduced a class of functions f ∈ A such that for 0 < μ < 1, He called it the class of function of non-Bazilevič type.There are many subordination results for this class (see [15]).In fact, this type of functions has been used to solve various problems (see [14]).
The main object of the present work is to apply a method based on the differential subordination in order to derive sufficient conditions for functions where q is a given univalent function in U such that q(z) = 0, μ = 0.Moreover, we give applications of these results in fractional calculus.We shall need the following known results: Lemma 1.1 ([4]).Let q(z) be univalent in the unit disk U and θ and φ be analytic in a domain D containing q(U ) with φ(w) = 0 when w ∈ q(U ).Set , then p(z) ≺ q(z) and q is the best dominant.

Subordination results.
In this section, we study subordination for normalized analytic functions in the classes A + α and A − α .Theorem 2.1.Let a function q be univalent in the unit disk U such that q(z) = 0, zq (z) q(z) is starlike univalent in U and (2.1) and q is the best dominant.
Proof.Let the function p be defined by By setting and By straightforward computation, we have Then by the assumption of the theorem, we see that the assertion of the theorem follows by application of Lemma 1.1.
Corollary 2.1.Assume that (2.1) holds and q is convex univalent in U .If Corollary 2.2.Assume that (2.1) holds and q is convex univalent in U .If Corollary 2.3.Assume that (2.1) holds and q is convex univalent in U .If and q(z) = e μAz is the best dominant.
The next result can be found in [3].
Corollary 2.4.Assume that k = 0 in Theorem 2.1.Then and q is the best dominant.Theorem 2.2.Let a function q(z) be convex univalent in the unit disk U such that q (z) = 0 and and q is the best dominant.
Proof.Let the function p be defined by By setting ψ = 1, it can easily be observed that Then by the assumption of the theorem we see that the assertion of the theorem follows by application of Lemma 1.2.

Corollary 2.5. Assume that (2.2) holds and
is the best dominant.
Corollary 2.6.Assume that (2.2) holds and q is convex univalent in U .If Corollary 2.7.Assume that (2.2) holds and q is convex univalent in U .If F ∈ A − α and and q(z) = e μAz is the best dominant.
The next result can be found in [3].
and q is the best dominant.

Applications.
In this section, we present some applications of Section 2 to fractional integral operators.Assume that f (z) = ∞ n=2 ϕ n z n−1 and let us begin with the following definitions: Definition 3.1 ([12]).The fractional integral of order α is defined, for a function f, by where the function f is analytic in a simply-connected region of the complex z-plane (C) containing the origin and the multiplicity of (z − ζ) α−1 is removed by requiring log(z − ζ) to be real when (z − ζ) > 0.

Thus we have
where a n := ϕnΓ(n) Γ(n+α) , for all n = 2, 3, . . . .This implies that z , so we get the following results: Theorem 3.1.Let the assumptions of Theorem 2.1 be satisfied.Then and q is the best dominant.
Proof.Consider the function F be defined by and q is the best dominant.
Proof.Consider the function F be defined by Let F (a, b; c; z) be the Gauss hypergeometric function (see [13]) defined, for z ∈ U, by where is the Pochhammer symbol defined by We need the following definition of fractional operators of the Saigo type fractional calculus (see [10], [9]).Definition 3.2.For α > 0 and β, η ∈ R, the fractional integral operator I α,β,η 0,z is defined by where the function f (z) is analytic in a simply-connected region of the zplane containing the origin, with the order From Definition 3.2, with β < 0, we have , so we have the following results: Theorem 3.3.Assume that the hypotheses of Theorem 2.1 are satisfied.Then and q is the best dominant.
Proof.Consider the function F defined by Theorem 3.4.Assume that the hypotheses of Theorem 2.2 are satisfied.Then and q is the best dominant.
Proof.Consider the function F defined by Remark 3.1.Note that the authors have recently studied and defined several other classes of analytic functions related to fractional power (see [2], [1], [4]).

The class S
To discuss our problem, we have to recall here the following lemma due to Jack [15].
where k is a real number and k ≥ 1.
We get the following result: Proof.Let w be defined by which contradicts the hypothesis (4.1).Therefore, we conclude that |w(z)| < 1 for all z ∈ U that is This completes the proof of the theorem.

1 .
Introduction and preliminaries.Consider the functions F in the open disk U := {z ∈ C : |z| < 1}, defined by and the multiplicity of (z − ζ) α−1 is removed by requiring log(z − ζ) to be real when z − ζ > 0.

Lemma 4 . 1 .
Let w be analytic in U with w(0) = 0.If |w(z)| attains its maximum value on the circle |z| = r < 1 at a point z 0 , then