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On uniformly univalent functions with respect to symmetrical points

Abstract

In this paper, we define and study some new subclasses of starlike and close-to-convex functions with respect to symmetrical points. These functions map the open unit disc onto certain conic regions in the right half plane. Some basic properties, a necessary condition, and coefficient and arc length problems are investigated. The mapping properties of the functions in these classes are studied under a certain linear operator.

MSC:30C45, 30C50.

1 Introduction

Let A be the class of functions of the form

f(z)=z+ n = 2 a n z n ,
(1.1)

which are analytic in the open unit disc E={z:|z|<1}. Let S, K, S , and C be the subclasses of A which consist of univalent, close-to-convex, starlike (with respect to origin), and convex functions, respectively. For recent developments, extensions, and applications, see [125] and the references therein.

A function f in A is said to be uniformly convex in E if f is a univalent convex function along with the property that, for every circular arc γ contained in E, with center ξ also in E, the image curve f(γ) is a convex arc. The class of uniformly convex functions is denoted by UCV. The corresponding class UST is defined by the relation that fUCV if, and only if, z f UST. It is well known [13] that fUCV if, and only if

| z f ( z ) f ( z ) |< { 1 + z f ( z ) f ( z ) } (zE).

Uniformly starlike and convex functions were first introduced by Goodman [3] and then studied by various other authors. If f,gA, we say f is subordinate to g in E, written as fg or f(z)g(z), if there exists a Schwarz function w(z) such that f(z)=g(w(z)) for zE.

For 0β<1, the class P(β) consists of functions p(z) analytic in E with p(0)=1 such that p(z)>β for zE, and, with β=0, we obtain the well-known class P of Carathéodory functions with positive real part.

For k[0,), the conic regions Ω k are defined as follows, see [5]:

Ω k = { u + i v : u > k ( u 1 ) 2 + v 2 } .

For fixed k, Ω k represents the conic regions bounded, successively, by the imaginary axis (k=0), the right branch of a hyperbolic (0<k<1) and a parabola v 2 =2u1 (k=1). When k>1, the domain becomes a bounded domain being the interior of the ellipse.

We shall consider the case when k[0,1]. Related to the domain Ω k , the following functions p k (z), k[0,1], play the role of extremal functions mapping in E onto Ω k :

p k (z)={ 1 + z 1 z ( k = 0 ) , 1 + 2 π 2 ( log 1 + z 1 z ) 2 ( k = 1 ) , 1 + 2 1 k 2 sinh 2 [ ( 2 π arccos k ) arctanh z ] ( 0 < k < 1 ) .
(1.2)

These functions are univalent in E and belong to the class P. Using the subordination concept, we define the class P( p k ) as follows.

Let p(z) be analytic in E with p(0)=1. Then pP( p k ) if, and only if, p p k in E and p k (z) are given by (1.2).

The conic domains Ω k can be generalized as given by

Ω k , β =(1β) Ω k +β,

with the corresponding extremal function

p k , β (z)=(1β) p k +β ( 0 β < 1 , k [ 0 , 1 ] ) .

It can easily be seen that the analytic function p(z), with p(0)=1, belongs to the class P( p k , β ) if p(z) p k , β (z) in E.

It is easy to verify that P( p k , β ) is a convex set. It is known [6] that

P( p k )P ( k k + 1 ) P,

and, for pP( p k ), we have

|argp(z)|σ π 2 ,

where

σ= 2 π arctan 1 k .
(1.3)

So we can write p(z)= h σ (z), hP.

Also

P( p k , β )P ( k + β k + 1 ) P.

Sakaguchi [24] introduced and studied the class S s of starlike functions with respect to symmetrical points. The class S s includes the classes of convex and odd starlike functions with respect to the origin. It was shown [24] that a necessary and sufficient condition for f S s to be univalent and starlike with respect to symmetrical points in E is that

( 2 z f ( z ) f ( z ) f ( z ) ) P,zE.

Das and Singh [2] defined the classes C s of convex functions with respect to symmetrical points and showed that a necessary and sufficient condition for f C s is that

2 ( z f ( z ) ) ( f ( z ) f ( z ) ) P,zE.

It is also well known [2] that f C s if, and only if, z f S s .

We now define the following.

Definition 1.1 Let fA. The f is said to be in the class kS T s (β) if, and only if,

2 z f ( z ) ( f ( z ) f ( z ) ) P( p k , β ),zE.

It can easily be seen that

kS T s (β) S s S s , β 1 = k + β k + 1 .

Also, for β=0=k, the class kS T s (β) reduces to S s .

The class kUC V s (β) is defined as follows.

Definition 1.2 Let fA. Then fkUC V s (β) if, and only if z f kS T s (β) for zE.

We note that

kUC V s (β) C s ( β 1 ) C s , β 1 = k + β k + 1 .

Definition 1.3 Let fA. Then fkU K s (β) if, and only if, there exists gkS T s (β) such that

( 2 z f ( z ) g ( z ) g ( z ) ) P( p k , β ),zE.

Since P( p k , β )P( β 1 )P, β 1 = k + β k + 1 , and kS T s (β) S s , we note that

kU K s (β) K s K,

where k S consists of close-to-convex functions with respect to symmetrical starlike functions.

From the definition, it is clear that kU K s (β) consists of univalent functions.

For k=0, β=0 and f(z)=g(z), kU K s (β) reduces to the class S s .

2 Preliminary results

We shall need the following lemmas to prove our main results.

Lemma 2.1 [15]

Let q(z) be a convex function in E with q(0)=1 and let another function h:EC be with h(z)>0. Let p(z) be analytic in E with p(0)=1 such that

( p ( z ) + h ( z ) z p ( z ) ) q(z),zE.

Then p(z)q(z), zE.

Lemma 2.2 Let N(z), D(z) be analytic in E with

N(0)=0=D(z)

and let D S for zE. Then N ( z ) D ( z ) P( p k , β ) implies that N ( z ) D ( z ) P( p k , β ) for zE.

Proof Let

N ( z ) D ( z ) =p(z).

Then

N ( z ) D ( z ) =p(z)+h(z) ( z p ( z ) ) ,h(z)= 1 h 0 ( z ) ,

where

h 0 (z)= z D ( z ) D ( z ) P.

Since N ( z ) D ( z ) P( p k , β ), we have

N ( z ) D ( z ) = ( p ( z ) + h ( z ) ( z p ( z ) ) ) p k , β (z),zE.

We now use Lemma 2.1 and this implies that

N ( z ) D ( z ) =p(z) p k , β (z)in E.

This proves that N ( z ) D ( z ) P( p k , β ) for zE. □

The following lemma is an easy extension of a result proved in [5].

Lemma 2.3 Let k[0,) and γ 1 , δ 1 be any complex numbers with γ 1 0 and let { γ 1 k k + 1 + δ 1 }>β. If h(z) is analytic in E, h(0)=1 and it satisfies

( h ( z ) + z h ( z ) γ 1 h ( z ) + δ 1 ) p k , β (z),
(2.1)

and q k , β (z) is an analytic solution of

( q k , β ( z ) + z q k , β ( z ) γ 1 q k , β ( z ) + δ 1 ) = p k , β (z),

then q k , β is univalent and

h(z) q k , β (z) p k , β (z),

and q k , β (z) is the best dominant of (2.1).

3 The class kS T s (β)

In this section, we shall study some basic properties of the class kS T s (β).

Theorem 3.1 Let fkS T s (β). Then the odd function

Ψ(z)= 1 2 [ f ( z ) f ( z ) ] ,
(3.1)

belongs to kST(β) in E.

In particular Ψ(z) is an odd starlike function of order β 1 = k + β k + 1 in E.

Proof Logarithmic differentiation of (3.1) and simple computation yield

z Ψ ( z ) Ψ ( z ) = 1 2 [ 2 z f ( z ) f ( z ) f ( z ) + 2 ( z ) f ( z ) f ( z ) f ( z ) ] = 1 2 [ p 1 ( z ) + p 2 ( z ) ] , for  z E , p 1 , p 2 P ( p k , β ) .

Since P( p k , β ) is a convex set, it follows that z Ψ ( z ) Ψ ( z ) P( p k , β ) and thus ΨkST(β) in E. □

As a special case, we note that, for k=0=β, 1 2 [f(z)f(z)]=Ψ(z) S in E, and hence z f Ψ P. We now discuss a geometric property for fkS T s (β). Here we investigate the behavior of the inclusion of the tangent at a point w(θ)=f(r e i θ ) to the image Γ r of the circle C r ={z:|z|=r}, 0r<1, θ[0,2π], under the mapping by means of a function f from the class fkS T s (β).

Let

Φ(θ)= π 2 +θ+arg f ( r e i θ ) =arg θ f ( r e i θ ) ,

and, for θ 2 > θ 1 , θ 1 , θ 2 [0,2π],

Φ( θ 2 )Φ( θ 1 )= θ 2 +arg f ( r e i θ 2 ) θ 1 arg f ( r e i θ 1 ) .

Now, since

θ+arg f ( r e i θ ) =θ+ { i ln f ( r e i θ ) } ,

then

θ ( θ + arg f ( r e i θ ) ) = { 1 + r e i θ f ( r e i θ ) f ( r e i θ ) } .

Hence

θ 1 θ 2 θ ( θ + arg f ( r e i θ ) ) dθ= θ 1 θ 2 { 1 + r e i θ f ( r e i θ ) f ( r e i θ ) } dθ.

Also, on the other hand,

θ 1 θ 2 θ ( θ + arg f ( r e i θ ) ) d θ = θ 2 + arg f ( r e i θ 2 ) θ 1 arg f ( r e i θ 1 ) = Φ ( θ 2 ) Φ ( θ 1 ) .

So, the integral on the left side of the last inequality characterizes the increment of the angle of the inclination of the tangent to the curve Γ r between the points w( θ 2 ) and w( θ 1 ) for θ 2 > θ 1 .

We have the following necessary condition for fkS T s (β).

Theorem 3.2 Let fkS T s (β). Then, with z=r e i θ and 0 θ 1 < θ 2 2π, 0β<1 and 0k1, we have

θ 1 θ 2 { ( z f ( z ) ) f ( z ) } dθ>σπ+2 cos 1 { 2 ( 1 β ) 1 ( 1 2 β ) r 2 } + β 1 ( θ 2 θ 1 ),

where σ is given by (1.3) and β 1 = k + β k + 1 .

Proof Since f ( z ) Ψ ( z ) P( p k , β ), Ψ(z)= 1 2 [f(z)f(z)] and ΨkUCV(β)C(β).

We can write

f (z)= ( Ψ 1 ( z ) ) 1 β 1 h σ (z), Ψ 1 C,hP(β),

and this gives us, with z=r e i θ , 0r<1, 0 θ 1 < θ 2 2π,

θ 1 θ 2 { ( z f ( z ) ) f ( z ) } d θ = ( 1 β 1 ) θ 1 θ 2 { ( z Ψ 1 ( z ) ) Ψ 1 ( z ) } d θ + σ θ 1 θ 2 2 h ( z ) h ( z ) d θ + β 1 ( θ 2 θ 1 ) .
(3.2)

For hP(β), we observe that

θ arg h ( r e i θ ) = θ { i ln h ( r e i θ ) } = { r e i θ h ( r e i θ ) h ( r e i θ ) } .

Therefore

θ 1 θ 2 { r e i θ h ( r e i θ ) h ( r e i θ ) } dθ=argh ( r e i θ 2 ) argh ( r e i θ 1 ) ,

and

max h P ( β ) | θ 1 θ 2 { r e i θ h ( r e i θ ) h ( r e i θ ) } dθ|= max h P ( β ) |argh ( r e i θ 2 ) argh ( r e i θ 1 ) |.

We can write

1 1 β [ h ( z ) β ] =p(z),pP,

and for |z|=r<1, it is well known that

|p(z) 1 + r 2 1 r 2 | 2 r 1 r 2 .

From this, we have

|h(z) 1 + ( 1 2 β ) r 2 1 r 2 | 2 ( 1 β ) r 1 r 2 .

Thus the values of h are contained in the circle of Apollonius whose diameter is the line segment from 1 ( 1 2 β ) r 1 + r to 1 + ( 1 2 β ) r 1 r and has the radius 2 ( 1 β ) r 1 r 2 . So |argh(z)| attains its maximum at points where a ray from origin is tangent to the circle, that is, when

argh(z)=± sin 1 ( 2 ( 1 β ) r 1 ( 1 2 β ) r 2 ) .
(3.3)

From (3.3), we observe that

max h P ( β ) | θ 1 θ 2 { r e i θ h ( r e i θ ) h ( r e i θ ) } d θ | 2 sin 1 ( 2 ( 1 β ) r 1 ( 1 2 β ) r 2 ) = π 2 cos 1 ( 2 ( 1 β ) r 1 ( 1 2 β ) r 2 ) .
(3.4)

Also, for Ψ 1 C,

θ 1 θ 2 { 1 + r e i θ Ψ 1 ( r e i θ ) Ψ 1 ( r e i θ ) } dθ0.
(3.5)

Using (3.4) and (3.5) in (3.2), we obtain the required result. □

We note the following special cases:

  1. 1.

    For k=0, 0 θ 1 < θ 2 2π, z=r e i θ , it follows from Theorem 3.2 that

    θ 1 θ 2 { 1 + z f ( z ) f ( z ) } dθ>π(zE).

This is a necessary and sufficient condition for f to be close-to-convex (hence univalent) in E; see [7]. This also shows that S T s (β)K.

  1. 2.

    For k=1 θ 1 θ 2 {1+ z f ( z ) f ( z ) }dθ> π 2 .

  2. 3.

    When k[0,1], it is obvious that σ(0,1]. In this case, the class kS T s (β) consists of strongly close-to-convex functions of order σ in the sense of Pommerenke [20, 21].

Theorem 3.3 (Integral representation)

Let fkS T s (β). Then

f (z)= 1 2 p(z)exp 0 z 1 t [ p ( t ) + p ( t ) 2 ] dt,

where pP( p k , β ), zE.

Proof Since fkS T s (β), we can write

2 z f ( z ) f ( z ) f ( z ) =p(z),pP( p k , β ).

This gives us

2 [ f ( z ) f ( z ) ] f ( z ) f ( z ) 1 z = 1 2 [ p ( z ) p ( z ) 2 ]

and the result follows when we integrate. □

When k=0, β=0, we obtain the result for the class S s given in [5].

We now study the class kS T s (β) under a certain integral operator.

Theorem 3.4 Let gkS T s (β) and let for m=1,2,3,,G be defined by

G(z)= m + 1 2 z m 0 z t m 1 { g ( t ) g ( t ) } dt.
(3.6)

Then G(z) belongs to kS T s (β) in E.

Proof Let

J(z)= 0 z t m 1 g ( t ) g ( t ) 2 dt.

Since gkS T s (β), 1 2 {g(z)g(z)}kST(β) S ( β 1 ) S , and β 1 = k + β k + 1 . Therefore it can easily be verified that J(z) is (m+1)-valently starlike in E.

We can write (3.6) as

z m G(z)=(m+1)J(z),

and, differentiating logarithmically, we have

z G ( z ) G ( z ) = z J ( z ) m J ( z ) J ( z ) = N ( z ) D ( z ) ,

say, where N(0)=D(0)=0 and D is (m+1)-valently starlike.

Let

N ( z ) D ( z ) =h(z).

Then

N ( z ) D ( z ) = h ( z ) + z h ( z ) h 0 ( z ) , h 0 ( z ) = z D ( z ) D ( z ) P = h ( z ) + H 0 ( z ) ( z h ( z ) ) , H 0 = 1 h 0 P .
(3.7)

Since

N ( z ) D ( z ) = ( z h ( z ) ) m J ( z ) J ( z ) = { ( z J ( z ) ) J ( z ) m } P ( p k , β ) .

We now apply Lemma 2.2 to obtain

N ( z ) D ( z ) = z G ( z ) G ( z ) P( p k , β ),zE.

This proves that GkST(β) in E. □

Theorem 3.5 Let f,gkS T s (β) and let F be defined by the following integral operator:

F(z)= ( γ + 1 δ ) z 1 1 δ 0 z t 1 δ 2 [ f ( t ) f ( t ) 2 ] 1 1 + γ [ g ( t ) g ( t ) 2 ] dt,
(3.8)

where zE, δ>0, γ0 and [ k ( 1 + γ ) k + 1 +( 1 δ 1)]>β. Then F(z) belongs to kST(β) for zE.

When g(z)=z, γ=0, we obtain a generalized form of the Bernardi operator; see [1]. Also for g(z)=z, γ=0, and δ= 1 2 , we have the well-known integral operator studied by Libera [11] who showed that it preserves the geometric properties of convexity, starlikeness, and close-to-convexity.

Proof Let f ( z ) f ( z ) 2 = Ψ 1 (z), g ( z ) g ( z ) 2 = Ψ 2 (z). Then Ψ 1 , Ψ 2 kST(β) in E. We can write (3.8) as

F(z)= ( γ + 1 δ ) z 1 1 δ 0 z t 1 δ 2 ( Ψ 1 ( t ) ) 1 1 + γ ( Ψ 2 ( t ) ) dt.
(3.9)

Differentiating (3.9) logarithmically, and with p(z)= z F ( z ) F ( z ) , we have

γ 1 + γ z Ψ 1 Ψ 1 ( z ) + 1 1 + γ z Ψ 2 Ψ 2 ( z ) =p(z)+ z p ( z ) ( 1 + γ ) p ( z ) + ( 1 δ 1 ) .
(3.10)

Since, for i=1,2, Ψ i kST(β), z Ψ 1 ( z ) Ψ 1 = h 1 (z), z Ψ 2 ( z ) Ψ 2 = h 2 (z) both belong to P( p k , β ) in E, and P( p k , β ) is a convex set. Therefore

( γ 1 + γ h 1 ( z ) + 1 1 + γ h 2 ( z ) ) P( p k , β ),zE.
(3.11)

From (3.10) and (3.11), it follows that

( p ( z ) + z p ( z ) ( 1 + γ ) p ( z ) + ( 1 δ 1 ) ) p k , β (z).

We now apply Lemma 2.3 which gives us

p(z) q k , β (z) p k , β (z).

Thus FkST(β) and the proof is complete. □

4 The class kU K s (β)

Here we shall study some properties of the class kU K s (β) which consists of k-uniformly close-to-convex functions.

Let L(r,f) denote the length of the image of the circle |z|=r under f. We prove the following.

Theorem 4.1 Let fkU K s (β). Then, for 0<r<1, k[0,1],

L(r,f)=O(1) ( 1 1 r ) σ β 1 , β 1 < σ 2 ,

where β 1 = k + β k + 1 and σ is given by (1.3), and O(1) is a constant depending only on k, β.

Proof For fkU K s (β), we can write

z f (z)=Ψ(z) h σ (z),hP,Ψ S ( β 1 ),
(4.1)

and Ψ(z)={g(z)g(z)}, gkS T s (β).

Since Ψ S ( β 1 ) and is odd, there exists an odd starlike function Ψ 1 (z) such that

Ψ(z)=z ( Ψ 1 ( z ) z ) 1 β 1 =z ( Ψ 1 ( z ) z ) 1 β 1 k + 1 .

Thus, with z=r e i θ ,

L(r,f)= 0 2 π |z f (z)|dθ= 0 2 π | z β 1 ( Ψ 1 ( z ) ) 1 β 1 h σ (z)|dθ,

and using Hölder’s inequality, we have

L(r,f)2π r β 1 ( 1 2 π 0 2 π | Ψ 1 ( z ) | ( 1 β ) ( z z σ ) d θ ) 2 σ z ( 1 2 π 0 2 π | h ( z ) | 2 d θ ) σ 2 .
(4.2)

For hP, it is well known [20] that

1 2 π 0 2 π |h(z) | 2 dθ 1 + 3 r 2 1 r 2 .
(4.3)

Using (4.3) and subordination for odd starlike functions in (4.2), it follows that

L ( r , f ) C ( β 1 , σ ) ( 1 1 r 2 ) [ ( 1 β 1 ) ( 2 2 σ ) 1 ] [ 1 + 3 r 2 1 r ] σ 2 = O ( 1 ) ( 1 1 r ) σ β 1 ,

where C and O(1) are constants depending only on β 1 and σ. This completes the proof. □

We now discuss the growth rate of coefficients of fkU K s (β).

Theorem 4.2 Let fkU K s (β) and be given by (1.1). Then

a n =O(1) n σ β 1 1 ,n1, β 1 < σ 2 ,

where O(1) is a constant depending only on σ and β 1 and σ, β 1 are as given in Theorem  4.1.

Proof For z=r e i θ , n1, Cauchy’s Theorem gives us

n | a n | = 1 2 π r n + 1 | 0 2 π z f ( z ) e i n θ d θ | 1 2 π r n + 1 0 2 π | z f ( z ) | d θ = 1 2 π r n L ( r , f ) .

With r=(1 1 n ), we use Theorem 4.1 and obtain the required result. □

Theorem 4.3 Let fkU K s (β) and let F be defined by

F(z)= m + 1 2 z m 0 z t m 1 { f ( t ) f ( t ) } dt.
(4.4)

Then FkU K s (β) in E. That is, the class kU K s (β) is preserved under the integral operator (4.4).

Proof Since fkU K s (β), we can write

{ 2 z f ( z ) g ( z ) g ( z ) } P( p k , β ),gkS T s (β) S S ( β 1 ).

Let G(z)= 1 2 { g 1 (z) g 1 (z)} and be defined by (3.5). By Theorem 3.4, g 1 kST(β) and Gk S s T(β) S s ( β 1 ). Let G=z G 1 . Then we can write

G 1 (z)= 1 2 [ z g 1 ( z ) g 1 ( z ) ] , G 1 kUC V s (β).

Thus, from (4.4) and g=z g 1 , g 1 C s ( β 1 ), we have

2 F ( z ) [ g 1 ( z ) g 1 ( z ) ] = z m { f ( z ) f ( z ) } m 0 z t m 1 { f ( t ) f ( t ) } d t z m { g 1 ( z ) g 1 ( z ) } m 0 z t m 1 { g 1 ( t ) g 1 ( t ) } d t = N ( z ) D ( z ) ,

say. We note that N(0)=D(0)=0, and for g 1 C S ( β 1 ),

( z D ( z ) ) D ( z ) = m + { z [ g 1 ( z ) g 1 ( z ) ] } { g 1 ( z ) g 1 ( z ) } = m + h 1 ( z ) , h 1 P ( β 1 ) .

Since P( β 1 ) is a convex set, D C s ( β 1 ) S in E. We thus have

N ( z ) D ( z ) = 1 2 [ 2 z f ( z ) [ g 1 ( z ) g 1 ( z ) ] + 2 ( z ) f ( z ) [ g 1 ( z ) g 1 ( z ) ] ] P( p k , β ).

Now, using Lemma 2.2, it follows that

N ( z ) D ( z ) = 2 F ( z ) ( g 1 ( z ) g 1 ( z ) ) P( p k , β )for zE.

This proves that FkU K S (β) in E. □

We study a partial converse of the above result as follows.

Theorem 4.4 Let ( 2 z f ( z ) g ( z ) g ( z ) ) p k (z) in E and let

F 1 (z)= 1 1 + m z 1 m ( z m f ( z ) ) ,m=1,2,3,.
(4.5)

Then F 1 K s for |z|< r 1 , where

r 1 = { m + 1 ( 2 β 1 ) + ( z β 1 ) 2 + ( m + 1 ) ( m 1 + 2 β 1 ) } , β 1 = k + β k + 1 .
(4.6)

Proof We shall need the following well-known results for pP(α), 0α<1; see [4]:

1 ( 1 2 α ) r 1 + r |p(z)| 1 + ( 1 2 α ) r 1 r ,
(4.7)
| p (z)| 2 [ p ( z ) α ] 1 r 2 .
(4.8)

Since fkU K s (β), there exists g S s ( β 1 ) such that, for zE.

( 2 z f ( z ) g ( z ) g ( z ) ) =p(z),pP( p k )P(α),α= k k + 1 .

From (4.5), we have

F 1 (z)= 1 1 + m [ m f ( z ) + z f ( z ) ] ,

and this gives us

2 z F 1 ( z ) g ( z ) g ( z ) = 1 m + 1 [ 2 m f ( z ) g ( z ) g ( z ) + 2 z ( z f ( z ) ) g ( z ) g ( z ) ] = 1 m + 1 [ m p ( z ) + z p ( z ) + p ( z ) h ( z ) ] ,

where

h(z)= z Ψ ( z ) Ψ ( z ) P( β 1 ),Ψ(z)=g(z)g(z).

Now, using (4.7) and (4.8), we have

{ 2 z F 1 ( z ) g ( z ) g ( z ) } ( p ( z ) α ) 1 + m { m + 1 ( 1 2 β 1 ) r 1 + r 2 r 1 r 2 } = p ( z ) α 1 + m [ T ( r ) 1 r 2 ] ,
(4.9)

where

T(r)=(m+1)2(2 β 1 )r+(m2 β 1 +1) r 2 .

We note that T(0)=1+m>0 and T(1)=3<0. So there exists r 1 (0,1). The right hand side of (4.9) is positive for |z|< r 1 , where r 1 is given by (4.6). This implies that F K s for |z|< r 1 and the proof is complete. □

We have the following special cases.

  1. 1.

    For k=0=β, f K s . Then F 1 , defined by (4.5) belongs to K s for |z|< r 0 = 1 + m 2 + 3 + m 2 .

  2. 2.

    When m=1 and β 1 =0 (that is, k=0=β), then F 1 (z)= ( z f ( z ) ) 2 belongs to the same class for |z|< 1 2 . This result has been proved by Livingston [12] for convex and starlike functions.

References

  1. Bernardi SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135: 429-446.

    Article  MathSciNet  MATH  Google Scholar 

  2. Das RN, Singh P: On subclasses of Schlicht mappings. Indian J. Pure Appl. Math. 1977, 8: 864-872.

    MathSciNet  MATH  Google Scholar 

  3. Goodman AW: On uniformly starlike functions. J. Math. Anal. Appl. 1991, 155: 364-370. 10.1016/0022-247X(91)90006-L

    Article  MathSciNet  MATH  Google Scholar 

  4. Goodman AW: Univalent Functions, Vol I & II. Polygonal Publishing House, Washington; 1983.

    MATH  Google Scholar 

  5. Kanas S: Differential subordination related to conic sections. J. Math. Anal. Appl. 2006,317(2):650-658. 10.1016/j.jmaa.2005.09.034

    Article  MathSciNet  MATH  Google Scholar 

  6. Kanas S: Techniques of differential subordination for domains bounded by conic sections. Int. J. Math. Math. Sci. 2003, 38: 2389-2400.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kanas S, Lecko A, Moleda A: Certain generalization of the Sakaguchi lemma. Folia Sci. Univ. Tech. Resov. 1987, 38: 35-42.

    MathSciNet  MATH  Google Scholar 

  8. Kanas S, Sugawa T: On conformal representation of the interior of an ellipse. Ann. Acad. Sci. Fenn., Math. 2006, 31: 329-348.

    MathSciNet  MATH  Google Scholar 

  9. Kanas S, Wisniowska A: Conic regions and k-uniform convexity. J. Comput. Appl. Math. 1999, 105: 327-336. 10.1016/S0377-0427(99)00018-7

    Article  MathSciNet  MATH  Google Scholar 

  10. Kaplan W: Close-to-convex Schlicht functions. Mich. Math. J. 1952, 1: 169-185.

    Article  MathSciNet  MATH  Google Scholar 

  11. Libera RJ: Some classes of regular univalent functions. Proc. Am. Math. Soc. 1965, 16: 755-758. 10.1090/S0002-9939-1965-0178131-2

    Article  MathSciNet  MATH  Google Scholar 

  12. Livingston AE: On the radius of univalence of certain analytic functions. Proc. Am. Math. Soc. 1966, 17: 352-359. 10.1090/S0002-9939-1966-0188423-X

    Article  MathSciNet  MATH  Google Scholar 

  13. Ma W, Minda D: Uniformly convex functions. Ann. Pol. Math. 1992,57(2):165-175.

    MathSciNet  MATH  Google Scholar 

  14. Ma W, Minda D: Uniformly convex functions II. Ann. Pol. Math. 1992,58(3):275-285.

    MathSciNet  MATH  Google Scholar 

  15. Miller SS, Mocanu PT: Univalent solution of Briot-Bouquet differential equations. J. Differ. Equ. 1985, 56: 297-308. 10.1016/0022-0396(85)90082-8

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor KI, Noor MA: Higher order close-to-convex functions related with conic domain. Appl. Math. Inf. Sci. 2014,8(5):2455-2463. 10.12785/amis/080541

    Article  MathSciNet  Google Scholar 

  17. Noor, KI, Ahmad, QZ, Noor, MA: On some subclasses of analytic functions defined by fractional derivative in the conic regions. Appl. Math. Inf. Sci. (2014/15)

  18. Noor KI, Fayyaz R, Noor MA: Some classes of k -uniformly functions with bounded radius rotation. Appl. Math. Inf. Sci. 2014,8(2):1-7.

    MathSciNet  Google Scholar 

  19. Noor, KI, Noor, MA, Murtaza, R: Inclusion properties with applications for certain subclasses of analytic functions. Appl. Math. Inf. Sci. (2014/15)

  20. Pommerenke C: On close-to-convex analytic functions. Trans. Am. Math. Soc. 1965, 14: 176-186.

    Article  MathSciNet  MATH  Google Scholar 

  21. Pommerenke C: On starlike and close-to-convex functions. Proc. Lond. Math. Soc. 1963, 3: 290-304.

    Article  MathSciNet  MATH  Google Scholar 

  22. Ronning F: On starlike functions associated with parabolic regions. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 1991, 45: 117-122.

    MathSciNet  MATH  Google Scholar 

  23. Ronning F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118: 189-196. 10.1090/S0002-9939-1993-1128729-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Sakaguchi K: On a certain univalent mapping. J. Math. Soc. Jpn. 1959, 11: 72-73. 10.2969/jmsj/01110072

    Article  MathSciNet  MATH  Google Scholar 

  25. Stankiewicz J: Some remarks on functions starlike with respect to symmetrical points. Ann. Univ. Mariae Curie-Skłodowska. Sect. A 1965, 19: 53-59.

    MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank editor and anonymous referee for their valuable suggestions. The author is grateful to Dr. SM Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing an excellent research and academic environment. This research is supported by HEC NRPU project No: 20-1966/R&D/11-2553, titled, Research unit od Academic Excellence in Geometric Function Theory and Applications.

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Noor, K.I. On uniformly univalent functions with respect to symmetrical points. J Inequal Appl 2014, 254 (2014). https://doi.org/10.1186/1029-242X-2014-254

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