Differential Subordinations of Arithmetic and Geometric Means of Some Functionals Related to a Sector

For r > 0 let r {z ∈ : |z| < r}. Let 1 . Let the functions f and F be analytic in the unit disc . A function f is called subordinate to F, written f ≺ F, if F is univalent in , f 0 F 0 and f ⊂ F . Let D be a domain in 2 and ψ : 2 ⊃ D → be an analytic function, and let p be a function analytic in with p z , zp′ z ∈ D, z ∈ and h be a function analytic and univalent in . The function p is said to satisfy the first-order differential subordination if


Introduction
For r > 0 let r {z ∈ : |z| < r}.Let 1 .Let the functions f and F be analytic in the unit disc .A function f is called subordinate to F, written f ≺ F, if F is univalent in , f 0 F 0 and f ⊂ F .Let D be a domain in 2 and ψ : 2 ⊃ D → be an analytic function, and let p be a function analytic in with p z , zp z ∈ D, z ∈ and h be a function analytic and univalent in .The function p is said to satisfy the first-order differential subordination if ψ p z , zp z ≺ h z , ψ p 0 , 0 h 0 , z ∈ .

1.1
The general theory of the differential subordinations has been studied intensively by many authors.A survey of this theory can by found in the monograph by Miller and Mocanu 1 .
For β ∈ 0, 2 let It is clear that h β maps univalently onto the sector of the angle βπ symmetrical with respect to the real axis with the vertex at the origin.
In this paper we are interested in the following problem referring to 1.1 to find the constant c k n, γ, α, β so that to the following relation is true: with suitable assumptions on function p and constants n, α, γ, β.For selected parameters n, γ, α, β the theorems presented here reduce to the well-known theorems proved by various authors.Particularly, results of this type can be applied to examine inclusion relation between subclasses of analytic functions defined with using arithmetic or geometric means of some functionals, for example, the class of α-convex functions or γ -starlike functions.
The lemma below that slightly generalizes a lemma proved by Miller and Mocanu 2 will be required in our investigation.Lemma 1.1 see 2 .Let q : → be a function analytic and univalent on , injective on ∂ and q 0 1.Let be analytic in , p / ≡ 1. Suppose that there exists a point z 0 ∈ such that p z 0 ∈ ∂q and If ξ 0 q −1 p z 0 and q ξ 0 exists, then there exists an m ≥ n for which z 0 p z 0 mξ 0 q ξ 0 .1.6

Main Results
In the first theorem which follows directly from Theorem 2.
. By H k n, α, γ will be denoted the class of functions p analytic in of the form 1.4 such that the function we see that Therefore the class H 1 n, α, 1 contains all analytic functions p of the form 1.4 .5 Let p be analytic function in of the form 1.4 .Suppose that p z 0 0 for some z 0 ∈ .Then where m ≥ 1 and p 1 is analytic function in with p 1 z 0 / 0 for z ∈ .Then we have .

2.10
Hence we see that for k 1 and γ ∈ 0, 1 or for k ≥ 2 and γ ∈ 0, 1 the function has a pole at z 0 .Therefore for such k and γ we see that every p ∈ H k n, α, γ is nonvanishing in D.
Theorem 2.5.Let k ∈ AE, α ≥ 0, γ ∈ 0, 1 and β ∈ 0, 1 be such that k Proof.The case α 0 is evident so we assume that α > 0. For k ∈ N and α > 0 let Φ Φ k,α be defined by 2.5 .For β ∈ 0, 1 the function Applying Theorem 2.2 with h β instead of h we get the assertion.Now we prove two theorems were we improve the result of Theorem 2.5.The problem 1.3 will be divided into two cases: k 1 and k > 1.
First we consider the case k 1.The theorem below was proved in 4 .To be selfcontained we include its proof.

2.18
Proof. 1 Assume that α > 0 and γ ∈ 0, 1 since the cases γ 0 or α 0 are evident.Suppose, on the contrary, that p is not subordinate to h β .Then, by the minimum principle for harmonic mappings there exists r 0 ∈ 0, 1 such that and one of the following cases hold: Therefore ξ 0 / ± 1 and Since ξ 0 / ± 1, so h β ξ 0 exists.Hence and by Lemma 1.1 there exists an m ≥ n for which

2.28
In view of the fact that x > 0 let us take

2.29
Hence and from 2.28 we have

2.30
By the above and by the fact that m ≥ n we have
Thus we arrive at a contradiction with 2.17 so p ≺ h β .4 When 2.21 holds, we see that x < 0 in 2.25 .Next we finish the proof by similar argumentations like in the above.
5 Assume now that 2.22 holds.In view of Remark 2.4 this is possible only when k γ 1. a For β < 1 the boundary ∂h β has the corner at 0 of the angle βπ < π.Since p ∂ r 0 is an analytic curve, in view of 2.19 the case p z 0 0 does not hold for β < 1.
Assume that p z 0 / 0. Since z 0 p z 0 is an outer normal to the curve p ∂ r 0 at p z 0 , by 2.19 we see that

International Journal of Mathematics and Mathematical Sciences
Hence taking into account that

2.35
we deduce that for all β ∈ 0, β 1 n, α, γ .In this way we arrive at a contradiction with 2.17 so p ≺ h β .
If p z 0 0, then and once again we contradict 2.17 .

Special Cases
1 The case n 1, α 1 was proved in 5 .
2 The case γ 1 was proved in 6 .

2.44
Proof. 1 We repeat argumentation from Parts 1 and 2 of the proof of Theorem 2.6. 2 We have

2.47
International Journal of Mathematics and Mathematical Sciences Thus, from 2.46 and by the fact that m ≥ n we obtain where We have Observe that the function a attains its minimum at the point

2.52
Hence, and from 2.48 , we have

2.54
Finally, the above and 2.53 yield Thus we arrive at a contradiction with 2.17 so p ≺ h β .4 For k − 1 β 1 we have c k n, α, γ, β β.This ends the proof of the theorem for the case x > 0. 5 When 2.21 holds, we see that x < 0 in 2.25 .Next we finish the proof by similar argumentations like in the above.
6 Since β ≤ 1/ k − 1 < 1, arguing as in Part 5 a of the proof of Theorem 2.6 we see that the case 2.22 does not hold.

Applications
All this type results can be applied in the theory of analytic functions.Some results concerning the inclusion relations between subclasses of analytic functions can be formulated.Let A n , n ∈ N, denote the class of functions of the form which is analytic in .For short, let A A 1 .Also let S denote the class of all functions in A which are univalent in .
To use theorems and corollaries listed in the previous section we put instead of the function p some functionals over the class A n , such as p z f z /z, p z zf z /f z or the others.In this way the inclusion relations between selected subclasses of analytic functions can be obtained.

Arithmetic Means
Using Corollary 2.7 we have the following.
The above result we can write in the following form.
Applying Theorem 2.6 with α 1 we have the following.
ii p z zf z /f z , z ∈ , f ∈ A n , n ∈ AE.For n ∈ AE, β ∈ 0, 1 , and γ ∈ 0, 1 let S * n γ, β denote class of functions f ∈ A n such that zf z f z where the operator L λ over the class A called Ruscheweyh derivative 24 was defined as follows: Remark 3.3. 1 The class T 1 1, 1 was introduced in 13 .2 The class T 1 α, 1 coincides with the class H α, 1, −1 studied in 14 .Observe that f ∈ T n α, β if and only if zf ∈ R n α, β .Let n ∈ AE, α ≥ 0, and β ∈ 0, β 1 n, α, 1 .If f ∈ A n and The class M 1 α18 ., that is, the class of so-called α-convex functions was introduced byMocanu 15.2The class M 1 0, 1 is identical with the class S * of starlike functions.The class M 1 1, 1 is identical with the class C of convex functions.3TheclassM 1 0, β denoted by S * β were defined by Brannan and Kirwan 16 and, independently, byStankiewicz 17,18 .Functions in this class are called strongly starlike of order β.The class M 1 1, β denoted by C * β contains functions called strongly convex of order β.Using Corollary 2.12 we have the following result proved by Marjono and Thomas 19 .Let n ∈ AE, α ≥ 0, and β Remark 3.13. 1 The class S * n γ, 1 , that is, the class of so-called γ -starlike functions was introduced by Lewandowski et al.21 .Using Theorem 2.8 we obtain results due to Darus and Thomas 22 .∈,f∈ A n , n ∈ AE, a > 0, δ ≥ 0, 3.30where the integral operator K δ a over the class A n was defined byKomatu 23as follows: Theorem 3.14.Let n ∈ AE, γ ∈ 0, 1 , and β ∈ 0, 1 .If f ∈ A n and