1. Introduction
Let H(U) be the set of functions which are regular in the unit disc U={z∈ℂ:|z|<1}, A={f∈H(U):f(0)=f′(0)-1=0} and S={f∈A:f is univalent in U}.
We recall here the definitions of the well-known classes of starlike and convex functions:
(1)S*={f∈A:Re(zf′(z)f(z))>0, z∈U},Sc={f∈A:Re(1+zf′′(z)f′(z))>0, z∈U}.
Let w be a fixed point in U and A(w)={f∈H(U):f(w)=f′(w)-1=0}.
In [1], Kanas and Rønning introduced the following classes:
(2)Sw={f∈A(w):f is univalent in U},STw=Sw*={f∈Sw:Re((z-w)f′(z)f(z))>0, z∈U},SVw=Swc={f∈Sw:1+Re((z-w)f′′(z)f′(z))>0, z∈U}.
These classes are extensively studied by Acu and Owa [2].
The class Sw* is defined by geometric property that the image of any circular arc centered at w is starlike with respect to f(w) and the corresponding class Swc is defined by the property that the image of any circular arc centered at w is convex. We observe that the definitions are somewhat similar to the ones for uniformly starlike and convex functions introduced by Goodman in [3, 4], except that in this case the point w is fixed.
In fact, several subclasses of S have been introduced by a fixed geometric property of the image domain. It is interesting to note that these subclasses play an important role in other branches of mathematics. For example, starlike functions and convex functions play an important role in the solution of certain differential equations (see Robertson [5], Saitoh [6], Owa et al. [7], Reade and Silverman [8], and Sokół and Wiśniowska-Wajnryb [9]). One of the important problems in geometric function theory are the extremal problems, which pose an effective method for establishing the existence of analytic functions with certain natural properties. Extremal problems (supremum Re{f}) play an important role in geometric function theory, for finding sharp estimates, coefficient bounds, and an extremal function. The results we obtained here may have potential application in other branches of mathematics, both pure and applied. For example the extremal problems are closely connected to Hele-Shaw flow of fluid mechanics [10], exterior inverse problems in potential theory, and many others.
Now let us begin with our definitions as follows.
The function f(z) in Sw is said to be starlike functions of order γ if and only if
(3)Re{(z-w)f′(z)f(z)}>γ (z∈U)
for some γ (0≤γ<1). We denote by STw(γ) the class of all starlike functions of order γ. Similarly, a function f(z) in Sw is said to be convex of order γ if and only if
(4)Re{1+(z-w)f′′(z)f′(z)}>γ (z∈U)
for some γ (0≤γ<1). We denote by SVw(γ) the class of all convex functions of order γ.
It is easy to see that Sw denoting the subclass of A(w) has the series of expansion:
(5)f(z)=(z-w)+∑n=2∞an(z-w)n.
For the function f(z) in the class Sw, we define the following new differential operator:
(6)T0f(z)=f(z),Tα,β,δ,λ1f(z)=(1-β(λ-α))f(z) +β(λ-α)(z-w)f′(z)+δ(z-w)2f′′(z),Tα,β,δ,λ2f(z)=(1-β(λ-α))(Tα,β,δ,λ1f(z)) +β(λ-α)(z-w)(Tα,β,δ,λ1f(z))′ +δ(z-w)2(Tα,β,δ,λ1f(z))′′,
and for m=1,2,3,…. (7)Tα,β,δ,λmf(z)=(1-β(λ-α))(Tα,β,δ,λm-1f(z)) +β(λ-α)(z-w)(Tα,β,δ,λm-1f(z))′ +δ(z-w)2(Tα,β,δ,λm-1f(z))′′=(z-w)+∑n=2∞[1+(n-1)(β(λ-α)+nδ)]m ×an(z-w)n,
for α≥0, β≥0, δ≥0, λ≥0, and m∈ℕ0=ℕ∪{0}.
It easily verified from (7) that
(8)β(λ-α)(z-w)(Tα,β,δ,λmf(z))′ =Tα,β,δ,λm+1f(z)-(1-β(λ-α))Tα,β,δ,λmf(z) -δ(z-w)2(Tα,β,δ,λmf(z))′′.
Remark 1.
(i) When w=δ=0, we have the operator introduced and studied by Darus and Ibrahim (see [11]).
(ii) When w=α=δ=0 and β=1, we have the operator introduced and studied by Al-Oboudi (see [12]).
(iii) When α=δ=0 and λ=β=1, we have the operator introduced and studied by Acu and Owa (see [2]).
(iv) And when α=δ=w=0 and λ=β=1, we have the operator introduced and studied by Sǎlǎgean (see [13]).
With the help of the differential operator Tα,β,δ,λm, we define the class Sm,w(α,β,γ,δ,λ) as follows.
Definition 2.
A function f(z)∈Sw is said to be a member of the class Sm,w(α,β,γ,δ,λ) if it satisfies
(9)|(z-w)(Tα,β,δ,λmf(z))′Tα,β,δ,λmf(z)-1| <|(z-w)(Tα,β,δ,λmf(z))′Tα,β,δ,λmf(z)+1-2γ|
for some 0≤γ<1, α≥0, β≥0, δ≥0, λ≥0, and m∈ℕ0 and for all z∈U.
It is easy to check that S0,0(0,1,γ,0,1) is the class of starlike functions of order γ and S0,0(0,1,0,0,1) gives the all class of starlike functions.
Let Tw denote the subclass of A(w) consisting of functions of the following form:
(10)f(z)=(z-w)-∑n=2∞an(z-w)n, an≥0.
Further, we define the class Sm,w*(α,β,γ,δ,λ) by
(11)Sm,w*(α,β,γ,δ,λ)=Sm,w(α,β,γ,δ,λ)∩Tw.
In this paper, coefficient inequalities, distortion theorem, and closure theorems of functions belonging to the class Sm,w*(α,β,γ,δ,λ) are obtained. Finally, the class preserving integral operators of the form
(12)F(z)=c+1(z-w)c∫wz(t-w)c-1f(t)dt (c>-1)
for the class Sm,w*(α,β,γ,δ,λ) is considered.
2. Coefficient Inequalities
Our first result provides a sufficient condition for a function, regular in U, to be in Sm,w*(α,β,γ,δ,λ).
Theorem 3.
A function f(z) being defined by (10) is in Sm,w*(α,β,γ,δ,λ) if and only if
(13)∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]man≤1-γ (m∈ℕ0),
where 0≤γ<1. The result (13) is sharp for functions of the following form:
(14)fn(z)=(z-w)-1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]m(z-w)n (n≥2;m∈ℕ0).
Proof.
Suppose that (13) holds true for 0≤γ<1. consider the expression
(15)M(f,f′)=|(z-w)(Tα,β,δ,λmf(z))′-(Tα,β,δ,λmf(z))| -|(z-w)(Tα,β,δ,λmf(z))′+(1-2γ)(Tα,β,δ,λmf(z))|.
Then for |z-w|=r<1, we have
(16)M(f,f′) =|∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞| -|∑n=2∞ 2(1-γ)(z-w) -∑n=2∞(n-2γ+1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞|,M(f,f′)≤∑n=2∞(n-1) ×[1+(n-1)β((λ-α)+nδ)]manrn -2(1-γ)+∑n=2∞(n+2γ-1) ×[1+λ(n+p-2)]manrn≤∑n=2∞2(n-γ)[1+(n-1)β((λ-α)+nδ)]m ×an-2(1-γ)≤0.
So that f(z)∈Sm,w*(α,β,γ,δ,λ).
For the converse, assume that
(17)|(z-w)(Tα,β,δ,λmf(z))′Tα,β,δ,λmf(z)-1| /|(z-w)(Tα,β,δ,λmf(z))′Tα,β,δ,λmf(z)+1-2γ| =|(an(z-w)n∑n=2∞)-1(-∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]m) ×(∑n=2∞2(1-γ)(z-w) -∑n=2∞(n-2γ+1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞)-1| <1.
Since Re(z-w)≤|z-w| for all (z-w), it follows from (17) that
(18)Re{(an(z-w)n∑n=2∞)-1(∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]m) ×(∑n=2∞2(1-γ)(z-w) -∑n=2∞(n-2γ+1)[1+(n-1)β((λ-α)+nδ)]m ×an(z-w)n∑n=2∞(n-2γ+1)[1+(n-1)β((λ-α)+nδ)]m)-1} <1.
Choose values of (z-w) on the real axis so that (z-w)(Tα,β,δ,λmf(z))′/Tα,β,δ,λmf(z) is real. Upon clearing the denominator in (18) and letting r→1- through real values, we obtain
(19)∑n=2∞(n-1)[1+(n-1)β((λ-α)+nδ)]man <2(1-γ) -∑n=2∞(n-2γ+1)[1+(n-1)β((λ-α)+nδ)]man.
This gives the required condition. Hence the theorem follows.
Corollary 4.
Let the function f(z) be defined by (10) and f(z)∈Sw. If f∈Sm,w*(α,β,γ,δ,λ), then
(20)an≤1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]m, n≥2.
The result (20) is sharp for functions fn(z) given by (14).
5. Closure Theorems
Let the functions fj(z), j=1,2,…,l, be defined by
(35)fj(z)=(z-w)-∑n=2∞an,j(z-w)n, (an,j≥0)
for z∈U.
Theorem 8.
Let the functions fj(z) be defined by (35) in the class Sm,w*(α,β,γ,δ,λ) for every j=1,2,…,l. Then the function G(z) defined by
(36)G(z)=(z-w)-∑n=2∞bn(z-w)n, (bn≥0)
is a member of the class Sm,w*(α,β,γ,δ,λ), where
(37)bn=1l∑j=1lan,j (n≥2).
Proof.
Since fj(z)∈Sm,w*(α,β,γ,δ,λ), it follows from Theorem 3 that
(38)∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]man,j≤1-γ
for every j=1,2,…,l. Hence,
(39)∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]mbn =∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]m ×{1l∑j=1lan,j} =1l∑j=1l(∑n=2∞(n-γ) ×[1+(n-1)β((λ-α)+nδ)]man,j1l∑j=1lan,j) ≤1l∑j=1l(1-γ)=1-γ
which (in view of Theorem 5) implies that G(z)∈Sm,w*(α,β,γ,δ,λ).
Theorem 9.
The class Sm,w*(α,β,γ,δ,λ) is closed under convex linear combination.
Proof.
Suppose that the functions fj(z) (j=1,2) defined by (35) be in the class Sm,w*(α,β,γ,δ,λ), it is sufficient to prove that the function
(40)H(z)=μf1(z)+(1-μ)f2(z) (0≤μ≤1)
is also in the class Sm,w*(α,β,γ,δ,λ). Since, for 0≤μ≤1,
(41)H(z)=(z-w)-∑n=2∞{μan,1+(1-μ)an,2}(z-w)n,
we observe that
(42)∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]m ×{μan,1+(1-μ)an,2} =μ∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]man,1 +(1-μ)∑n=2∞(n-γ) ×[1+(n-1)β((λ-α)+nδ)]man,2 ≤μ(1-γ)+(1-μ)(1-γ)=(1-γ).
Hence H(z)∈Sm,w*(α,β,γ,δ,λ). This completes the proof of Theorem 9.
Theorem 10.
Let
(43)f0(z)=z-w,fn(z)=(z-w)-1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]m(z-w)n (n≥2;m∈ℕ0).
Then f(z)∈Sm,w*(α,β,γ,δ,λ) if and only if it can be expressed in the form
(44)f(z)=∑n=1∞λnfn(z),
where λn≥0 and ∑n=1∞λn=1.
Proof.
Suppose that
(45)f(z)=∑n=1∞λnfn(z),
where λn≥0 (n≥1) and ∑n=1∞λn=1. Then
(46)f(z)=∑n=1∞λnfn(z)=λ1f1(z)+∑n=2∞λnfn(z)=(z-w) -∑n=2∞1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]m ×λn(z-w)n.
Since
(47)∑n=2∞(n-γ)[1+(n-1)β((λ-α)+nδ)]m ·1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]mλn =(1-γ)∑n=2∞λn =(1-γ)(1-λ0)≤1-γ,
it follows from Theorem 3 that the function f(z)∈Sm,w*(α,β,γ,δ,λ). Conversely, let us suppose that f(z)∈Sm,w*(α,β,γ,δ,λ). Since
(48)an≤1-γ(n-γ)[1+(n-1)β((λ-α)+nδ)]m (n≥2;m∈ℕ0).
Setting
(49)λn=(n-γ)[1+(n-1)β((λ-α)+nδ)]m1-γan, (n≥2;m∈ℕ0),λ1=1-∑n=2∞λn,
it follows that f(z)=Sm,w*(α,β,γ,δ,λ). This completes the proof of the theorem.