New Applications of the Bernardi Integral Operator

: Let A ( p , n ) be the class of f ( z ) which are analytic p -valent functions in the closed unit disk U = { z ∈ C : | z | ≤ 1 } . The expression B − m − λ f ( z ) is deﬁned by using fractional integrals of order λ for f ( z ) ∈ A ( p , n ) . When m = 1 and λ = 0, B − 1 f ( z ) becomes Bernardi integral operator. Using the fractional integral B − m − λ f ( z ) , the subclass T p , n ( α s , β , ρ ; m , λ ) of A ( p , n ) is introduced. In the present paper, we discuss some interesting properties for f ( z ) concerning with the class T p , n ( α s , β , ρ ; m , λ ) . Also, some interesting examples for our results will be considered.

For B −1 f (z) in (2), we consider and B −m f (z) = B −1 (B −m+1 f (z)) = z p + ∞ ∑ k=p+n p + γ k + γ m a k z k (6) with m ∈ N and B 0 f (z) = f (z). From the various definitions of fractional calculus of f (z) ∈ A(p, n) (that is, fractional integrals and fractional derivatives) given in the literature, we would like to recall here the following definitions for fractional calculus which were used by Owa [3] and Owa and Srivastava [4].
where the multiplicity of (z − t) λ−1 is removed by requiring log(z − t) to be real when z − t > 0 and Γ is the Gamma function.
With the above definitions, we know that for λ > 0 and f (z) ∈ A(p, n). Using the fractional integral operator over A(p, n), we consider where 0 ≤ λ ≤ 1. If λ = 0 in (9), then B 0 f (z) = f (z) and if λ = 1 in (9), then we see that With the operator B −λ f (z) given by (9), we know where 0 ≤ λ ≤ 1 and m ∈ N. The operator B −m−λ f (z) is a generalization of the Bernardi integral operator From s different boundary points z l (l = 1, 2, 3, ..., s) with |z l | = 1, we consider where for some real ρ > 0, we say that the function f (z) belongs to the class T p,n (α s , β, ρ; m, λ) .
It is clear that a function f (z) ∈ A(p, n) belongs to the class T p,n (α s , β, ρ; m, λ) provided that the condition is satisfied. If we consider the function f (z) ∈ A(p, n) given by Therefore, f (z) given by (16) is in the class T p,n (α s , β, ρ; m, λ) . Discussing our problems for f (z) ∈ T p,n (α s , β, ρ; m, λ) , we have to recall here the following lemma due to Miller and Mocanu [5,6] (refining the old one in Jack [7].) Lemma 1. Let the function w(z) given by w(z) = a n z n + a n+1 z n+1 + a n+2 z n+2 + . . . , n ∈ N (18) be analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a point z 0 , (0 < |z 0 | < 1) then there exists a real number k ≥ n such that and

Properties of Functions Concerning with the Class T p,n (α s , β, ρ; m, λ)
We begin with a sufficient condition on a function f (z) ∈ A(p, n) which makes it a member of T p,n (α s , β, ρ; m, λ) .
for some α s given by (13) with α s = 1 such that z g ∈ ∂U (g = 1, 2, 3, ..., s), and for some real ρ > 1, then Proof. We introduce the function w(z) defined by Noting that we obtain that and that by employing (21). Assume, to arrive at a contradiction, that there exists a point z 0 , Then, we can write that w( Since this contradicts our condition (21), we see that there is no z 0 , (0 < |z 0 | < 1) such that |w(z 0 )| = ρ > 1. This shows us that that is, that This completes the proof of the theorem.

Example 1.
We consider a function f (z) ∈ A(p, n) given by For such f (z), we have Now, we consider five boundary points such that and For these five boundary points, we know that and Thus α 5 is given by Q|a p+n |.
This gives us that Q|a p+n | with β = 0. For such α 5 and β, we take ρ > 1 with n p+γ Q|a p+n It follows from the above that For such α 5 and ρ > 1, f (z) satisfies Our next result reads as follows.
Theorem 3. If f (z) ∈ A(p, n) satisfies for some α s defined by (13) with α s = 1, g = 1, 2, 3, ..., m, and for some real ρ > 1, then Proof. Define the function w(z) by It follows from the above that By the definition of B −m−λ+g f (z), we know that

Example 3.
Consider the function f (z) = z p + a p+n z p+n , z ∈ U (76) which satisfies It follows from (77) that where Q is given by (33). Now, we consider the five boundary points z 1 , z 2 , z 3 , z 4 and z 5 as in Example 1.
Then we see Q|a p+n | where β = 0. With the above relation (79), we consider ρ > 1 such that that is, ρ satisfies Thus, we have that Remark 1. If we take γ = 1 in the results of this section, then these results correspond to applications of the Libera integral operator as introduced by Libera [2]. Let us write that for γ = 1 in (11). Then Theorem 1 says that if f (z) ∈ A(p, n) satisfies for some α s given by where − π 2 ≤ β ≤ π 2 , and for some real ρ > 1, then For another result, we consider again the Libera integral operator with γ = 1.

Application of Carathéodory Lemma
In this section, we will apply Carathéodory Lemma for coefficients of functions f (z) ∈ A(p, n).
The inequality (88) is sharp for each k.
Applying the above lemma, we derive the following thorem.