STARLIKE AND CONVEX FUNCTIONS WITH ...

ao

DOI: 10.1515/ms-2017-0083 Math. Slovaca 68 (2018), No. 1, 89–102

O PY

STARLIKE AND CONVEX FUNCTIONS WITH RESPECT TO SYMMETRIC CONJUGATE POINTS INVOLVING CONICAL DOMAIN C. Ramachandran* — R. Ambrose Prabhu** — Srikandan Sivasubramanian*** (Communicated by Stanislawa Kanas )

C

ABSTRACT. Enough attentions to domains related to conical sections has not been done so far although it deserves more. Making use of the conical domain the authors have defined a new class of starlike and Convex Functions with respect to symmetric points involving the conical domain. Growth and distortion estimates are studied with convolution using domains bounded by conic regions. Certain coefficient estimates are obtained for domains bounded by conical region. Finally interesting application of the results are also highlighted for the function Ωk,β defined by Noor.

R

c

2018 Mathematical Institute Slovak Academy of Sciences

1. Introduction

O

Let A denote the class of all functions f of the form: ∞ X f (z) = z + an z n

(1.1)

TH

n=2

U

which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}. Also let S, S ∗ (α), C(α) and K(α) denote the subclasses of A consisting of functions which are univalent, starlike of order α, convex of order α and close to convex of order α in U. In particular, the classes S ∗ (0) = S ∗ , C(0) = C and K(0) = K are the familiar classes of starlike and convex functions in U. ∞ ∞ P P For f (z) = z + an z n and g(z) = z + bn z n , then the Hadamard product (or convolution) n=2

n=2

A

is given by

(f ∗ g)(z) = z +

∞ X

an bn z n .

n=2

For two functions f and g analytic in U, we say f is subordinate to g denoted by f ≺ g, if there exists a Schwarz function w, analytic in U with w(0) = 0 and |w(z)| < 1, z ∈ U such that f (z) = g(w(z)), z ∈ U. A function f in A is said to be uniformly convex in U if f is a univalent convex function along with property that, for every circular arc γ contained in U, with center ξ also in U, the image curve f (γ) is a convex arc. The class of uniformly convex function is denoted by UCV. Uniformly starlike and convex functions were first introduced by Goodman [3] and Ronning [31, 32].

2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary: 30C45, 30C55, 30C80. K e y w o r d s: analytic function, univalent function, differential subordination, Hadamard product and conical region.

89

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN

Goodman [3,4] stated the following two variable analytic characterization of the classes denoted respectively by UCV and UST Theorem

1.1. A function f ∈ A is in UCV if and only if   (z − ζ)f 00 (z) < 1+ > 0, f 0 (z)

z∈U

for all z and ζ are in U.

1.2. A function f ∈ A is in UST if and only if   f (z) − f (ζ) > 0, z∈U < (z − ζ)f 0 (z)

O PY

Theorem

for all z and ζ are in U.

However, the classical Alexander theorem f ∈ C if and only if zf 0 ∈ S ∗ does not hold between the classes UCV and UST . A one variable characterization for UCV was obtained by A. W. Goodman [3, 4], Ma and Minda [17] and Ronning. [31]. The concepts of UCV and UST were extended to k − UCV and k − ST as follows: 1.3. A function f ∈ A is in k − U CV if and only if 00   zf (z) zf 00 (z) < 1+ 0 ≥ k 0 (z ∈ U) f (z) f (z)

C

Theorem

1.4. A function f ∈ A is in k − ST if and only if 0  0  zf (z) zf (z) − 1 (z ∈ U) < ≥ k f (z) f (z)

R

Definition

U

TH

O

The class k − UCV was introduced by Kanas and Wi´sniowska [12], where its geometric definition and connections with the conic domains were considered. The class k −ST was investigated in [13]. In fact, it is related to the class k−UCV by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions (see also the work of Kanas and Srivastava [11] for further developments involving each of the classes k −UCV and k −ST ). In particular, when k = 1, k − UCV ≡ U CV and k − ST = SP, was obtained, where UCV and SP are the familiar classes of uniformly convex functions and parabolic starlike functions in U respectively. It is remarked here that the classes k − UCV ≡ UCV and, i.e., k − ST = SP, are related to the domain bounded by conical sections. Recently Ramachandran et al. [29] studied the domain Ωk introduced by Kanas and Wi´sniowska [12] defined as follows: Ωk = {u + iv : u2 > k 2 (u − 1)2 + k 2 v 2 }.

(1.2)

A

For fixed k, the above domain represents the conic region bounded, successively, by the imaginary axis (k = 0), the right branch of a hyperbola (0 < k < 1), a parabola (k = 1), and an ellipse (k > 1). Also it was noted that, for no choice of parameter k (k > 1), Ωk reduces to a disk (see Figure. 1). The univalent functions mapping the unit disk U onto Ωk , denoted by qk , such that qk (0) = 1 and qk0 (0) > 0 are presented as follows: 1+z 1−z  √  2 1+ z 2 √ qk (z) = 1 + 2 log π 1− z √
2 qk (z) = 1 + sinh2 A(k) arctan h z 2 1−k qk (z) =

90

(z ∈ U), k = 0. (z ∈ U), k = 1. (z ∈ U), 0 < k < 1

4

C. Ramachandran, R.Ambrose Prabhu and S.Sivasubramanian

STARLIKE AND CONVEX FUNCTIONS

v

0π0 are presented

k 21−+ 1z

sin

2κ(t)

F √ ,t t

C

qk (z) = 1 +

(z ∈ U), k > 1

(z ∈ U), k = 0. 1−z  the√Legendre  Here, A(k) = π2 arccos k, while F(ω, t) is elliptic integral of the first kind 2 1+ z √ qk (z) = 1 + 2ωlog2 (z ∈ U), k = 1. πZ 1− z dx 2√ 2 √ arctan h, √z
κ(t) = F(1, F(ω, t) = qk (z) = 1 + sinh A(k) (z ∈ U) , 0 < t) k 1 πκ0 (t) t k −k1 = cosh 2κ(t) and t ∈ (0, 1) is chosen such that which maps the unit disk U onto

R

qk (z) =

4κ(t)

U

for k > 1 ,

By virtue of

A

O

the conic

TH

domains are

respectively for= 02 < k 1 , 2 2 2 1 −kx 2 1 − t x √ 1   0 1−k 1−k2 0 πκ (t) and t ∈ (0, 1) is chosen such that k = cosh which maps the unit disk U 4κ(t)   !2 ! onto the conic domains are respectively for2 0 < k < 1, k2   v 2    u + k22−1 2  < 1 + Ωk =  u + iv : k  1   . k    u +k21−1  √vk2 −1 2 − k     Ωk = u + iv :   −  > 1 , 1 k   √ k − ST can definition of k −  be reframed subordination, the U CV and 2 2 1−k 1−k zf 0 (z) p(z) = ≺ qk (z) f (z)

or

as

zf 00 (z) p(z) = 1 + 0 ≺ qk (z), f (z)

and the property of the domains are

k . k+1 A function f ∈ A is starlike with respect to symmetric points in U if, for every r close to 1, r < 1 and every z0 on |z| = r, the angular velocity of f (z) about f (−z0 ) is positive at z = z0 as z traverse the circle |z| = r, in the positive orientation. This class has been introduced and studied by Sakaguchi [33] and also, recently studied by Noor [21]–[24]. They have proved that the condition is equivalent to   zf 0 (z) >0 z ∈ U. < f (z) − f (−z)

91

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN

A function f ∈ A is starlike with respect to conjugate points in U if f satisfies the condition   zf 0 (z) < >0 z ∈ U. f (z) + f¯(¯ z)

O PY

A function f ∈ A is starlike with respect to symmetric conjugate points in U if f satisfies the condition   zf 0 (z) >0 z ∈ U, < f (z) − f¯(−¯ z) ∗ respectively. These classes which denote the classes consisting of these functions by Sc∗ and Ssc have been introduced by El-Ashwah and Thomas [2]. The functions in these classes are close to convex functions, and hence, it is univalent. Sokol [35] has introduced two more parameter in this class and obtained structural formula, the coefficient estimate, the radius of convexity and results about the neighbourhoods of functions (see also Sokol [36]). Padmanaban [25] has introduced the neighbourhood of functions in the class UST . Parvatham and Premabai [26] have introduced the following class of functions k − ST s :   zf 0 (z) zf 0 (z) < ≥ k − 1 (z ∈ U). (1.3) f (z) − f (−z) f (z) − f (−z)

Definition

C

Motivated essentially by the work of Kanas [6]–[14] and [20], in the current study the following classes have been defined: 1.5. A function f ∈ A is in the class Ss∗ (qk ) if

Definition

R

2zf 0 (z) ≺ qk (z). f (z) − f (−z)

1.6. A function f ∈ A is in the class Cs (qk ) if

O

2(zf 0 (z))0 ≺ qk (z). f 0 (z) + f 0 (−z)

TH

∗ (qk ) denote the corresponding classes of starlike functions with respect to Let Sc∗ (qk ) and Ssc conjugate points and symmetric conjugate points, respectively. The functions kqk n (n = 2, 3, . . . ) defined by kqk n (0) = kq0 k n (0) − 1 = 0 and

2zkq0 k n (z) + kq00k n (z) = qk (z n−1 ), kq0 k n (z) + kq0 k n (−z)

A

U

are examples of functions functions in Cqk . The functions hqk n satisfying zkq0 k n (z) = hqk n are examples of functions in Sq∗k .

2. Fekete Szeg˝ o inequality using conical domains

2.1. Let qk be the Riemann map of U onto Ω satisfying qk (0) = 1 and qk0 (0) = 0. We define the class of functions P(qk ) as follows:

Definition

P(qk ) = {h ∈ A : h(z) ≺ qk (z)}.

(2.1)

The geometry of the region Ωk and the explicit form of the function qk with basic properties in the class P(qk ) were discussed in detail by Kanas [6]–[14], [15]. In the particular case k = 0, P(q0 ) has been discussed as the familiar class of Caratheodary functions h ∈ A satisfying h(0) = 1 and 0 for z ∈ U. Also, Salma Faraj Ramadan [30] and Mishra [19] have considered the Fekete 92

STARLIKE AND CONVEX FUNCTIONS

Szeg¨ o coefficient bounds for typical classes of univalent functions. For the class of Caratheodory functions, i.e., P(q0 ), the following result is well known: Theorem

2.2 ([1]). Let the function h ∈ A satisfying h(0) = 1 and 0 for z ∈ U. If (z ∈ U),

h(z) = 1 + b1 z + b2 z 2 + . . .

   2 + (u − 1)|b1 |2 , u >     1 |b2 − ub21 | ≤ 2 − |b1 |2 , u=  2      2 − u|b1 |2 , u≤

1 ; 2 1 ; 2 1 . 2

O PY

then for −∞ < u < ∞.

(2.2)

(2.3)

Indeed, Kanas [15] has obtained similar estimates for the class P(qk ) (0 ≤ k < ∞) for the domain bounded by conical region as follows: Theorem 2.3 ([15]). Let 0 ≤ k < ∞ be fixed and the function qk of Definition 1.1 be represented by the Taylor-Maclaurin series

C

qk (z) = 1 + Q1 (k)z + Q2 (k)z 2 + · · ·

If the function h given by the Taylor-Maclaurin series (2.2) is a for −∞ < u < ∞,    Q1 (k) + (u − 1)Q1 (k)2 ,     2 |b2 − ub1 | ≤ Q1 (k),       Q1 (k) − uQ1 (k)2 ,

(z ∈ U).

member of the class P(qk ), then

1 ; 2 1 u= ; 2 1 u≤ . 2 The middle inequality in (2.5) is sharp for the function h(z) = qk (z 2 ). u>

(2.5)

TH

O

R

(2.4)

A

U

Also, the coefficient namely Q1 (k) and Q2 (k) may be written as follows: Let k ∈ [0, ∞) be fixed and qk be the function as in Definition 1.5. If the equation (2.4), then  2   2A , 0 ≤ k < 1;    1 − k2   8 , k = 1; (2.6) Q1 = Q1 (k) = π2     π2   √ , k > 1.  2 2 4κ (t)(k − 1)(1 + t) t

and

where

A=

2 π

Q2 = Q2 (k) = D(k)Q1 (k),

 2 A +2   , 0 ≤ k < 1;    3   8 , k = 1; D = D(k) = π2     (4κ2 (t)(t2 + 6t + 1) − π 2 )   √ , k > 1.  24κ2 (t)(1 + t) t

(2.7)

(2.8)

arccos k and κ(t) is the complete elliptic integral of first kind.

93

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN Lemma

2.4 ([16]). If the function ω ∈ B0 = {ω ∈ A : ω(0) = 0, |ω(z)| < 1, z ∈ U} is (z ∈ U).

ω(z) = d1 z + d2 z 2 + · · ·

(2.9)

Then for every complex number s, |d2 − sd21 | ≤ 1 + (|s| − 1)|d1 |2 .

(2.10)

The following lemma has been introduced by Prokhorov and Szynal [27], also studied by Ramachandran et al. [28]: 2.5 ([27]). If ω ∈ Ω, for any real numbers q1 and q2 , the following sharp estimate holds:

O PY

Lemma

|ω3 + q1 ω1 ω2 + q2 ω13 | ≤ H(q1 , q2 ),

where

for (q1 , q2 ) ∈ D1 ∪ D2 ,

for (q1 , q2 ) ∈ ∪7k=3 Dk ,

for (q1 , q2 ) ∈ D8 ∪ D9 ,

for (q1 , q2 ) ∈ D10 ∪ D11 r {±2, 1},

C

 1,       |q2 , |        21  |q1 | + 1  2 (|q1 | + 1) , H(q1 , q2 ) = 3 3(|q1 | + 1 + q2 )  1     q12 − 4 2 q2 q12 − 4    ,  2  3 q1 − 4q2 3(q2 − 1)     21   2 |q1 | − 1   (|q1 | − 1) , 3 3(|q1 | − 1 − q2 )

(2.11)

for (q1 , q2 ) ∈ D12 .

R

(2.12)

The extremal functions, up to a rotation, are of the form ω(z) = z 3 ,

ω(z) = z,

O

z[(1 − λ)ε2 + λε1 ] − ε1 ε2 1 − [(1 − λ)ε1 + λε2 ]z, z z(t2 + z) z(t1 − z) , ω(z) = ω2 (z) = , ω(z) = ω1 (z) = 1 − t1 z 1 + t2 z

TH

ω(z) = ω0 (z) =

|ε1 | = |ε2 | = 1,

ε1 = t0 − e r

−iθ0 2

(a ∓ b),

ε2 = −e

−iθ0 2

(ia ∓ b),

θ0 θ0 b±a , b = 1 − t20 sin2 , λ= , 2 2 2b h i 12 h 2q (q 2 + 2) − 3q 2 i 21 |q1 | + 1 2 1 1 , t1 = t0 = 2 3(q2 − 1)(q1 − 4q2 ) 3(|q1 | + 1 + q2 ) 1 h i |q1 | − 1 q1 h q2 (q12 + 8) − 2(q12 + 2) i θ0 2 = , cos t2 = 3(|q1 | − 1 − q2 ) 2 2 2q2 (q12 + 2) − 3q12

A

U

a = t0 cos

While the sets Dk , k = 1, 2, . . . , 12, are defined as follows: n o 1 D1 = (q1 , q2 ) : |q1 | ≤ , |q2 | ≤ 1 , 2 n o 1 4 D2 = (q1 , q2 ) : ≤ |q1 | ≤ 2, (|q1 | + 1)3 − (|q1 | + 1) ≤ q2 ≤ 1 , 2 27 n o 1 D3 = (q1 , q2 ) : |q1 | ≤ , |q2 | ≤ −1 , 2 n o 1 2 D4 = (q1 , q2 ) : |q1 | ≥ , |q2 | ≤ − (|q1 | + 1) , 2 3

94

STARLIKE AND CONVEX FUNCTIONS

D5 = {(q1 , q2 ) : |q1 | ≤ 2, |q2 | ≥ 1} ,

C

O PY

n o 1 2 D6 = (q1 , q2 ) : 2 ≤ |q1 | ≤ 4, |q2 | ≥ (q1 + 8) , 12 n o 2 D7 = (q1 , q2 ) : |q1 | ≥ 4, |q2 | ≥ (|q1 | − 1) , 3 o n 2 4 1 (|q1 | + 1)3 − (|q1 | + 1) , D8 = (q1 , q2 ) : ≤ |q1 | ≤ 2, − (|q1 | + 1) ≤ q2 ≤ 2 3 27   2 2|q1 |(|q1 + 1|) D9 = (q1 , q2 ) : |q1 | ≥ 2, − (|q1 | + 1) ≤ q2 ≤ 2 , 3 q1 + 2|q1 | + 4   1 2 2|q1 |(|q1 + 1|) ≤ q2 ≤ D10 = (q1 , q2 ) : 2 ≤ |q1 | ≤ 4, 2 (q + 8) , q1 + 2|q1 | + 4 12 1   2|q1 |(|q1 − 1|) 2|q1 |(|q1 + 1|) ≤ q2 ≤ 2 , D11 = (q1 , q2 ) : |q1 | ≥ 4, 2 q1 + 2|q1 | + 4 q1 − 2|q1 | + 4   2|q1 |(|q1 − 1|) 2 D12 = (q1 , q2 ) : |q1 | ≥ 4, 2 ≤ q2 ≤ (|q1 | − 1) . q1 − 2|q1 | + 4 3

2.6. Let 0 ≤ k < ∞ be fixed and the function qk of Definition 1.6 be represented by (2.2). If the function h given by the Taylor-Maclaurin series (2.2) is a member of the function class P(qk ), then for −∞ < µ < ∞.  2   µQ1 (k) − 2Q2 (k) ,  µ > α1 (k);   4   Q1 (k) (2.13) |a3 − µa22 | ≤ , α2 (k) ≤ µ ≤ α1 (k);  2     2Q2 (k) − µQ21 (k)   , µ < α2 (k). 4

U

where

TH

O

R

Theorem

α1 (k) =

2(D(k) + 1) , Q1 (k)

(2.14)

α2 (k) =

2(D(k) − 1) , Q1 (k)

(2.15)

A

and Q1 (k),Q2 (k) and D(k) are given by (2.6)–(2.8), respectively. All the inequality in (2.13) are sharp. Further, Q1 (k) H(q1 , q2 ) (2.16) |a4 | ≤ 3 where H(q1 , q2 ) is as defined in Lemma 2.5, 2Q2 (k) 1 + , Q1 (k) 2 Q3 (k) Q2 (k) q2 (k) = + . Q1 (k) 2 q1 (k) =

(2.17)

These results are sharp. 95

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN

P r o o f. From the definition of subordination, there exists a function ω ∈ B0 such that h(z) =

2zf 0 (z) = qk (ω(z)) f (z) − f (−z)

(z ∈ U).

ω and qk are given by the series (2.4) and (2.9) respectively, and a simple calculation yields Q1 (k)d1 , 2 Q1 (k)d2 + Q2 (k)d21 Q1 (k) a3 = = {d2 + D(k)d21 }, 2 2 Q1 (k)  a4 = d3 + q1 d1 d2 + q2 d31 , 4 where q1 and q2 as defined in (2.17). Therefore, a3 − µa22 =

O PY

a2 =

µ Q1 (k) [d2 + {D(k) − Q1 (k)}d21 ]. 2 2

This gives

(2.18)

Q1 (k) µ |d2 − d21 + {1 + D(k) − Q1 (k)}d21 |. 2 2 Let µ > α1 (k), then using the estimates |d2 −d21 | ≤ 1 from Lemma 2.4 and the well known estimates |d1 | ≤ 1 of the Schwarz lemma, we get

C

|a3 − µa22 | =

R

µ µQ21 (k) − 2Q2 (k) Q1 (k) [1 + {−1 − D(k) + Q1 (k)}] = (2.19) 2 2 4 This gives the first inequality in (2.13). On the other hand, if µ < α2 (k), then (2.18) gives |a3 − µa22 | ≤

Q1 (k) µ [1 + {|d2 | + D(k) − Q1 (k)|d21 }]. 2 2 2 Applying the estimates |d2 | ≤ 1 − |d1 | of Lemma 2.4 and |d1 | ≤ 1,the inequality

µ 2Q2 (k) − µQ21 (k) Q1 (k) [1 + {D(k) − Q1 (k) − 1}|d1 |2 ] = 2 2 4

TH

|a3 − µa22 | ≤

O

|a3 − µa22 | ≤

(2.20)

U

is achieved. This is the last inequality in (2.13). Lastly, if α2 (k) ≤ µ ≤ α1 (k), then (2.18) gives |D(k) − 2µQ1 (k)| ≤ 1. Therefore, Q1 (k) Q1 (k) Q1 (k) [|d2 + |d1 |2 ] ≤ [1 − |d1 |2 + |d1 |2 ] = 2 2 2 hence, the middle inequality in (2.13), is achieved. The sharpness of the inequality in (2.13) has been discussed next. Suppose µ > α1 (k). Then equality holds in (2.13), i.e., in (2.19) if d21 = −1 and |d2 − d21 | = 1. Therefore, ω(z) = iz and the extremal function is qk (iz). Next, if µ < α2 (k), then equality holds in (2.18) gives, i.e., in (2.20) if d1 = 1 and |d2 | = 1. Therefore, ω(z) is a rotation of z and the extremal function is rotation of qk (z). Lastly, if α2 (k) ≤ µ ≤ α1 (k), then equality holds in (2.18) if d1 = 0 and |d2 | = 1. Therefore, ω(z) is a rotation of z 2 and h(z) = qk (eiθ z 2 ). 

A

|a3 − µa22 | ≤

Remark 2.7. In the paper [34] Shanmugam et al. have proved similar bounds in the class φ. For instance, when µ ≤ σ1 , the authors obtained the estimate |a3 − µa22 | ≤ 12 [B2 − µ2 B12 ]. Observe that in the case of p(qk ) the result is far better; the same holds in the remaining range of the constant µ. 96

STARLIKE AND CONVEX FUNCTIONS

Further,in the current paper, it is observed that, for the choice of µ = 0, the coefficient bound as |a3 | ≤ 1 and for µ = 1, the coefficient bound as |a3 − a22 | ≤ Q12(k)

3. Convolution theorem

and Ωk =



O PY

In this section, a characterization of the class Cs (qk ) in terms of convolution has been presented. Denote by Ωk with 0 ≤ k < ∞, the following sets:   2(zf 0 (z))0 : z ∈ U, f ∈ C (q ) Ωk = s k f 0 (z) + f 0 (−z) 2zf 0 (z) : z ∈ U, f ∈ Ss∗ (qk ) f (z) − f (−z)



Note that, by (1.2), Ωk is a domain such that, 1 ∈ Ωk and ∂Ωk is a curve defined by the equality,

C

∂Ωk = {w : w = u + iv : u2 = k 2 (u − 1)2 + k 2 v 2 },

(0 ≤ k < ∞).

3.1. Let 0 ≤ k < ∞. A function f ∈ S is in the class Cs (qk ) if and only if z1 (f ∗Gt )(z) 6= 0 in U for all t ≥ 0, such that t2 − (kt − 1)2 ≥ 0, where    1 − z 2  1+z z 1 , z∈U − C(t) 1 + Gt (z) = 1 − C(t) (1 − z)2 1 − z 1+z p and C(t) = kt ± i t2 − (kt − 1)2 .

O

R

Theorem

P r o o f. Let 0 ≤ k < ∞. For a function f ∈ S, let define p(z) = p(0) = 1, it follows that

2(zf 0 (z))0 f 0 (z)+f 0 (−z) ,

z ∈ U. Since

U

TH

f ∈ Cs (qk ) ⇐⇒ p(z) ∈ / ∂Ωk for all z ∈ U. p Note that ∂Ωk = C(t) = kt ± i t2 − (kt − 1)2 , where t(k + 1) ≥ 1 and (1 − k)t ≥ −1. Also, note that     1 − z 2  z z z ∗ f (z) = 1+ + ∗ f (z) = zf 0 (z) + zf 0 (−z) (1 − z)2 (1 + z)2 (1 − z)2 1+z

A

and

Hence,

Thus,

  0  z z(1 + z) ∗ f (z) = zf 0 (z) + z 2 f 00 (z). ∗ f (z) = z (1 − z)3 (1 − z)2

1 1 (f ∗ Gt )(z) = (f 0 (z) + zf 00 (z) − C(t)[f 0 (z) + f 0 (−z)]) z 1 − C(t)  f 0 (z) + f 0 (−z)  (zf 0 (z))0 = − C(t) . 1 − C(t) f 0 (z) + f 0 (−z) 1 (f ∗ Gt )(z) 6= 0 z

⇐⇒

p(z) ∈ / ∂Ωk

⇐⇒

p(z) ∈ Ωk ,

z ∈ U.  97

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN

3.2. Let 0 ≤ k < ∞. A function f ∈ S is in the class Ss∗ (qk ) if and only if z1 (f ∗Gt )(z) 6= 0 in U for all t ≥ 0, such that t2 − (kt − 1)2 ≥ 0, where h 1 i z 1−z 1 C(t) , z∈U − Gt (z) = 1 − C(t) (1 − z)2 1 + z (1 + z)2 p and C(t) = kt ± i t2 − (kt − 1)2 . Theorem

0

and

 −z   z  +f ∗ = f (z) − f (−z). 1−z 1+z

C

f∗

O PY

2(zf (z)) P r o o f. Let 0 ≤ k < ∞. For a function f ∈ S, let define p(z) = f (z)−f (−z) , z ∈ U. Since p(0) = 1, it follows that / ∂Ωk for all z ∈ U. f ∈ Ss∗ (qk ) ⇐⇒ p(z) ∈ p 2 2 Note that ∂Ωk = C(t) = kt ± i t − (kt − 1) , where t(k + 1) ≥ 1 and (1 − k)t ≥ −1. Also, note that     1 + z 2  z z z ∗ f (z) = 1 − ∗ f (z) = zf (z) − zf (−z) − (1 − z)2 (1 + z)2 (1 − z)2 1−z

Hence,

Thus,

O

1 (f ∗ Gt )(z) 6= 0 z

R

1 1 (f ∗ Gt )(z) = (zf (z) − C(t)f (z)) z 1 − C(t) f (z) [z − C(t)]. = 1 − C(t) p(z) ∈ / ∂Ωk

⇐⇒

p(z) ∈ Ωk ,

z ∈ U. 

TH

⇐⇒

4. Growth, distortion and covering theorem

U

Assume that the function qk (z) is an analytic function with positive real part in the unit disk U, and qk (U) is convex and symmetric with respect to the real axis, qk (0) = 1 and qk0 (o) > 0. The functions kqk n (z) (n = 2, 3, . . . ) denoted by kqk n (0) = kq0 k n (0) − 1 = 0 and

A

2zkq0 k n (z) + kq00k n (z) = qk (z n−1 ) kq0 k n (z) + kq0 k n (−z)

are important examples of functions in C(qk ). The functions hqk n satisfying zkq0 k n (z) = hqk n are examples of functions in S ∗ (qk ). Write kqk 2 simply as kqk and hqk 2 simply as hqk . Theorem

(1)

4.1 ([18]). Let min |φ(z)| = φ(−r), max |φ(z)| = φ(r), |z| = r. If f ∈ C(φ), then

kφ0 (−r)

|z|=r

0

≤ |f (z)| ≤

kφ0 (r)

(2) kφ (−r) ≤ |f (z)| ≤ kφ (r)

(3) f (U) ⊃ {w : |w| ≤ −kφ (−1)}.

The result is sharp. 98

|z|=r

STARLIKE AND CONVEX FUNCTIONS

If f (z) = z + ak+1 z k+1 + · · · ∈ C(qk ), then we can prove that (see [5]) 1

1

[kφ0 (−rk )] k ≤ |f 0 (z)| ≤ [kφ0 (rk )] k . We prove the following Theorem

(1)

4.2. Let min |qk (z)| = qk (−r), max |qk (z)| = qk (r), |z| = r. If f ∈ Cc (qk ), then |z|=r

kq0 k (−r)

0

≤ |f (z)| ≤

|z|=r

kq0 k (r)

(2) −kqk (−r) ≤ |f (z)| ≤ kqk (r)

O PY

(3) f (U) ⊃ {w : |w| ≤ −kqk (−1)}.

The result is sharp.

P r o o f. Since f ∈ Cs (qk ) and qk is convex and symmetric with respect to real axis, it follows that f (z)+f¯(¯ z) is in C(qk ). Since g ∈ C(qk ), it follows that g 0 (z) ≺ kq0 k (z). Now, 2 (rkq0 k (−r))0 = kq0 k (−r) − rkq00k (−r) ≤ kq0 k (−r)qk (r) ≤ |(zf 0 (z))0 |,

C

(zf 0 (z))0 0 g (z) ≤ qk (r)kq0 k (r) = kq0 k (r) + rkq00k (r) ≤ (rkq0 k (r))0 . g 0 (z)

|(zf 0 (z))0 | =

By integrating from 0 to r, it follows that

kq0 k (−r) ≤ |f 0 (z)| ≤ kq0 k (r).

R

The second part follows from the first one, since −kqk (−r) is increasing in (0, 1) and bounded by 1. Here, −kqk (−1) = lim −kqk (−r). The results are sharp for the function f (z) = kqk (z) ∈ Cc (qk ), r→1

Theorem



4.3. Let min |qk (z)| = qk (−r), max |qk (z)| = qk (r), |z| = r. If f ∈ Sc∗ (qk ), then |z|=r

h0qk (−r)

0

≤ |f (z)| ≤

|z|=r

h0qk (r)

TH

(1)

O

since it has real coefficients and is in C(qk ).

(2) −hqk (−r) ≤ |f (z)| ≤ hqk (r)

(3) f (U) ⊃ {w : |w| ≤ −hqk (−1)}. The result is sharp.

A

U

P r o o f. The first part follows from above theorem and the fact zf 0 ∈ Sc∗ (qk ) if and only if f ∈ Cc (qk ). Let zf 0 (z) 2zf 0 (z) = p(z) = g(z) f (z) + f¯(¯ z)

where g(z) =

f (z)+f¯(¯ z) . 2

Since g ∈ S ∗ (qk ), and hence, −hqk (−r) ≤ |f (z)| ≤ hqk (r).

Therefore, for |z| = r < 1, h0qk (−r)

qk (−r)hqk (−r) g(z) qk (r)hqk (r) ≤ p(z) = h0qk (r) = = |f 0 (z)| ≤ −r z r

which represents the second part of our result. The other part follows easily.



Similar theorems are true for the classes of functions with respect to symmetric conjugate points. 99

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN Theorem

4.4. Let min |qk (z)| = qk (−r), max |qk (z)| = qk (r), |z| = r. If f ∈ Cs (qk ), then |z|=r

1 r

Zr

|z|=r

1 qk (−r)[kq0 k (−r2 )] 2 dr

1 ≤ |f (z)| ≤ r 0

0

Zr

1

qk (r)[kq0 k (r2 )] 2 dr

0

The other results for this class may be obtained easily and hence omitted. [f (z)−f (−z)] 2

= z + a3 z 3 + · · · ∈ C(qk ). Then the result follows 

O PY

P r o o f. The function g(z) = easily.

5. Application

C

With rotation and translation, the generalized form of the conical domain Ωk defined by Noor [20] is given by Ωk,β = (1 − β)Ωk + β, with the corresponding extremal function pk,β (z) = (1 − β)pk + β (0 ≤ β < 1, k ∈ [0, 1]). If p(z) ≺ pk,β (z) belongs to the class P (pk,β ), the following theorem associated with Fekete Szeg˝o coefficient bounds will be easily verified. 5.1. Let 0 ≤ k < ∞ be fixed and the function qk of Definition 1.6 be represented by (2.2). If the function h given by the Taylor-Maclaurin series (2.2) is a member of the function class P(qk ), then for −∞ < µ < ∞,  µ(1 − β)2 Q21 (k) − 2(1 − β)Q2 (k)   , µ > α1 (k);   4  (1 − β)Q1 (k) |a3 − µa22 | ≤ (5.1) , α2 (k) ≤ µ ≤ α1 (k);  2   2 2   2(1 − β)Q2 (k) − µ(1 − β) Q1 (k) , µ < α2 (k). 4 where 2(D(k) + 1) α1 (k) = , (5.2) (1 − β)Q1 (k) 2(D(k) − 1) α2 (k) = , (5.3) (1 − β)Q1 (k)

TH

O

R

Theorem

A

U

and Q1 (k),Q2 (k) and D(k) are given by (2.6)–(2.8) respectively. All the inequality in (5.1) are sharp. Further (1 − β)Q1 (k) H(q1 , q2 ) (5.4) |a4 | ≤ 3 where H(q1 , q2 ) is as defined in Lemma 2.5, 2(1 − β)Q2 (k) 1 + , (1 − β)Q1 (k) 2 Q3 (k) (1 − β)Q2 (k) q2 (k) = + . Q1 (k) 2 q1 (k) =

(5.5)

These results are sharp. Acknowledgement. The work of the third author is supported by a grant from Department of Science and Technology, Government of India vid ref: SR/FTP/MS-022/2012 under fast track scheme. 100

STARLIKE AND CONVEX FUNCTIONS REFERENCES

A

U

TH

O

R

C

O PY

[1] DUREN, P. L.: Univalent Functions. Grundlehren Math. Wiss., Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [2] EL-ASHWAH, R. M.—THOMAS, D. K.: Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc. 2 (1987), 85–100. [3] GOODMAN, A. W.: On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92. [4] GOODMAN, A. W.: On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364–370. [5] GRAHAM, I.—VAROLIN, D.: Bloch constants in one and several variables, Pacific J. Math. 174 (1996), 347–357. [6] KANAS, S.: Alternative characterization of the class k-U CV and related classes of univalent functions, Serdica Math. J. 25 (1999), 341–350. [7] KANAS, S.: Techniques of the differential subordination for domains bounded by conic sections, Internat. J. Math. Sci. 38 (2003), 2389–2400. [8] KANAS, S.: Differential subordination related to conic sections, J. Math. Anal. Appl. 317 (2006), 650–658. [9] KANAS, S.: Subordination for domains bounded by conic sections, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), 589–598. [10] KANAS, S.: Norm of pre-Schwarzian derivative for the class of k-uniform convex and k-starlike functions, Appl. Math. Comput. 215 (2009), 2275–2282. [11] KANAS, S.—SRIVASTAVA, H. M.: Linear operators associated with k-uniform convex functions, Integral Transforms Spec. Funct. 9 (2000), 121–132. ´ [12] KANAS, S.—WISNIOWSKA, A.: Conic regions and k-uniform convexity, II, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 22 (1998), 65–78. ´ [13] KANAS, S.—WISNIOWSKA, A.: Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327–336. ´ [14] KANAS, S.—WISNIOWSKA, A.: Conic regions and k-starlike function, Rev. Roum. Math. Pures Appl. 45 (2000), 647–657. [15] KANAS, S.: Coefficient estimates in subclasses of the Carath´ eodory class related to conic domains, Acta Math. Univ. Comenian. 74 (2005), 149–161. [16] KEOGH, F. R.—MERKES, E. P.: A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. [17] MA, W. C.—MINDA, D.: Uniformly convex functions, I, II, Ann. Polon. Math. 57 (1992), 165–175. [18] MA, W. C.—MINDA, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Internat. Press. Cambridge, MA., (Tianjin, 1992), pp. 157–169. [19] MISHRA, A. K.—GOCCHAYAT, P.: Fekete-Szego problem for k-uniformly convex functions and for a class defined by the Owa-Srivastava operator, J. Math. Anal. Appl. 347 (2008), 563–572. [20] NOOR, K. I.: On uniformly univalent functions with respect to symmetrical points, J. Inequal. Appl. (2014). [21] NOOR, K. I.—NOOR, M. A.: Higher order close-to-convex functions related with conic domain, Appl. Math. Inf. Sci. 8 (2014), 2455–2463. [22] NOOR, K. I.—AHMAD, Q. Z.—NOOR, M. A.: On some subclasses of analytic functions defined by fractional derivative in the conic regions, Appl. Math. Inf. Sci. 15 (2014). [23] NOOR, K. I.—FAYYAZ, R.—NOOR, M. A.: Some classes of k-uniformly functions with bounded radius rotation, Appl. Math. Inf. Sci. 8 (2014), 1–7. [24] NOOR, K. I.—NOOR, M. A.—MURTAZA, R.: Inclusion properties with applications for certain subclasses of analytic functions, Appl. Math. Inf. Sci. 15 (2014). [25] PADMANABHAN, K. S.: On uniformly convex functions in the unit disk, J. Anal. 2 (1994), 87–96. [26] PARVATHAM, R.—PREMABAI, M.: Neighbourhoods of uniformly starlike functions with respect to symmetric points, Bull. Calcutta. Math. Soc. 88 (1996), 155–160. [27] PROKHOROV, D. V.—SZYNAL, J.: Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae CurieSklodowska, Sect. A 35 (1984), 125–143. [28] RAMACHANDRAN, C.—SIVASUBRAMANIAN, S.—SILVERMAN, H.: Certain coefficient bounds for p-valent functions, Int. J. Math. Sci. 2007, Article ID 46576. [29] RAMACHANDRAN, C.—ANNAMALAI S.—SIVASUBRAMANIAN, S.: Inclusion relations for Bessel functions for domains bounded by conical domains, Adv. Difference Equ. (2014), 288. [30] RAMADAN, S. F.—DARUS, M.: On the Fekete-Szeg¨ o ineqality for a class of analytic functions defined by using generalized differential operator, Acta Univ. Apulensis Math. Inform. 26 (2011), 167–178. [31] RONNING, F. L.: Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196.

101

C. RAMACHANDRAN — R. AMBROSE PRABHU — SRIKANDAN SIVASUBRAMANIAN [32] RONNING, F.: Integral representations of bounded starlike functions, Ann. Polon. Math. 60 (1995), 289–297. [33] SAKAGUCHI, K.: On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72–75. [34] SHANMUGAM, T. N.—RAMACHANDRAN, C.—RAVICHANDRAN, V.: Fekete Szeg¨ o problem for subclasses of starlike functions with respect to symmetric points, Bull. Korean Math. Soc. 43 (2006), 589–598. [35] SOKOL, J.: Function starlike with respect to conjugate points, Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz. 12 (1991), 53–64. [36] SOKOL, J.—SZPILA, A.—SZPILA M.: On some subclass of starlike functions with respect to symmetric points, Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz. 12 (1991), 65–73. Received 3. 10. 2015 Accepted 15. 5. 2016

O PY

* Department of Mathematics University College of Engineering Villupuram Anna University, Villupuram 605 103 INDIA E-mail: [email protected] ** Department of Mathematics College of Engineering Guindy Anna University, Chennai 600 025, Tamilnadu INDIA E-mail: [email protected]

A

U

TH

O

R

C

*** Department of Mathematics University College of Engineering Tindivanam Anna University, Tindivanam 604 001 INDIA E-mail: [email protected]

102