Let U:=z:z∈ℂ and |z|<1 be the open unit disk and H(U) denote the class of all analytic functions f defined in the open unit disc U. For a positive integer n and a∈ℂ, let

We define the class A by

Let P denote the class of Carathéodory functions p such that p(0)=1,Rep(z)>0 and

The Möbius function L0(z)=1+z1−z,z∈U or its rotation acts as an extremal function for the class P and maps the open unit disc onto the right half-plane. Recall also the class P(γ)⊂P, 0≤γ<1, consisting of functions p∈P such that ℜ(p(z))>γ in U. For f,g∈H(U), we say that the function f is subordinate to the function g and write f(z)≺g(z), if there exists a Schwarz function w, that is, w∈H(U), with w(0)=0 and |w(z)|<1, such that f(z)=g(w(z)). For a univalent function g, f(z)≺g(z) if and only if f(0)=g(0) and f(U)⊂g(U). For reference, see [10]. Applying subordination, Janowski [6] introduced the class P[A,B] for −1≤B<A≤1. A function p analytic in U such that p(0)=1 belongs to the class P[A,B], if

where w is a Schwarz function. Geometrically, the image p(U) of p∈P[A,B] lies inside the open unit disc centered on the real axis with diameter ends at p(−1)  and p(1). Clearly, P[A,B]⊂P1−A1−B. The class P[A,B] is related with the class P of functions with positive real parts by p∈P if and only if

The simplest representation of the conic domain Δk is

This domain represents the right half plane for k=0, a hyperbola for 0<k<1, a parabola for k=1 and an ellipse for k>1. A normalized analytic function f is close-to-convex, if and only if there exists a function g∈C such that

We denote the class ofclose-to-convex functions by K. This class was introduced by Kaplan [16]. Sakaguchi [13] defined the class of starlike functions with respect to symmetric points as follows:

Definition 1.1. Let f∈S. Then f∈SCP, if it satisfies the condition:

For f∈A, f∈CCP [18] if and only if zf′∈SCP, where CCP represents the class of convex functions with respect to conjugate points. Various authors studied the class CCP of functions convex with respect toconjugate points and its subclasses, for detail, see [11, 12, 16, 21]. Obviously, it is a subclass of close-to-convex functions and hence it represents univalent functions. Moreover, this class includes the class of convex functions and odd starlike functions with respect to the origin, see [13].

A normalized analytic function f∈Q the class of quasi convex, if and only if zf′∈K. Silverman [19] introduced the class T of analytic univalent functions f having non-negative coefficients in the series representation:

The class TS∗γ a subclasses of T is defined by TS∗γ=S∗γ∩T. Selvaraj and Selvakumaran [14] investigated the classes SCP(δ,k,γ) and CCP(δ,k,γ) defined below:

where

and

0≤δ≤1,0≤γ<1,k≥0. Here we let  TSCP(δ,k,γ)=T∩SCP(δ,k,γ). A function f∈CCP(δ,k,γ) [18] if and only if zf′ ∈SCP(δ,k,γ). For δ=0, we have the classes SCP(k,γ) and CCP(k,γ) and also the classes

These and related classes have been introduced by various authors, for example, see [2, 3, 4, 5, 7, 9, 15, 20].

Definition 1.2. A function f∈KCP(δ,k,γ,β) if we have

where Fδ is given in (1.5),Gz=g(z)+g(z¯)¯:g∈TCCP(k,γ) and the class TCCP(k,γ) is defined in (1.6), 0≤δ≤1,0≤γ<1,k≥0 and 0≤β<1.

Here we let

where T is defined by (1.3). In our investigation of the class TKCP(δ,k,γ,β), we obtain necessary and sufficient conditions, coefficient estimates, distortion bounds, extreme points, radii of close-to-convexity, starlikeness and convexity. We also obtain integral means inequality and other related properties.

We now state some useful lemmas which we may need to establish our results in the sequel.

Lemma 2.1. If γ∈R and ω∈C then

Lemma 2.2. If γ∈R and ω∈C then

Lemma 2.3. ([14]) Let f∈TCCP(k,γ). Then

Lemma 2.4. ([14]) Let f,g∈S with f≺g. Then for η>0 and z=reiθ, we have

We employ the similar techniqueof Aqlan et al. [1] to find the coefficient estimates for the functions in the class TKCP(δ,k,γ,β).

Theorem 3.1. A function f∈TKCP(δ,k,γ,β), if and only if it satisfies the coefficient conditions

for 0≤δ≤1,0≤γ<1,k≥0,0≤β<1 and bn is given by

Proof. Suppose that f∈A satisfies the condition given in (3.1). We prove that f∈TKCP(δ,k,γ,β). In view of Lemma 2.1, it is enough to show that

where Fδ is given by (1.5), Gz=gz−g−z, and g∈CCP(δ,k,γ). We now consider that

where

To obtain (3.2) or (3.3), in view of Lemma 2.1, we need to prove that

Also for H and J as above, we see that

Again we consider that

or we can write

Keeping in view (3.6),(3.7) and (3.8), we have

For the limiting value of r, that is, r→1  and in view of (3.1) and (3.9), we obtain

Again this inequlity alongwith (3.1) implies (3.6). Hence f∈TKCP(δ,k,γ,β). Conversely, suppose that f∈TKCP(δ,k,γ,β). By Lemma 2.2, we have (3.2), i.e.,

or

or

and

Using (3.11) and (3.12) in (3.10), we get

We choose z=r>0 to have

For Re−eiθ>−eiθ=−1 and θn=1−δ+nδ, the above inequality becomes

Again taking r→1−, we get

where

This completes the proof.

As an immediate consequence of the above Theorem 3.1, we have the following theorem for the coefficients estimates of  f∈TKCP(δ,k,γ,β).

Theorem 3.2. For a function f:f∈TKCP(δ,k,γ,β), we have

for 0≤δ≤1,0≤γ<1,k≥0,0≤β<1.

Proof. For f∈TKCP(δ,k,γ,β), we use Theorem 3.1 to have

Using Lemma 2.3 in (3.13), we have

This implies that

The equality holds for the function

Now we solve the growth problems for functions in the class TKCP(δ,k,γ,β).

Theorem 4.1.If a function f∈TKCP(δ,k,γ,β), then for z=rexpiθ∈U, we have

and

Proof. For f∈TKCP(δ,k,γ,β), the inequality (3.1) yields

This implies that

Thus from fz≤z+z2∑ n=2∞an with 0<z=r<1, we have

Similarly for the lower bounds on the left side of (4.1), we can write

The inequalities (4.3) and (4.4) lead to (4.1). On the same lines, we can find the lower and upper bounds of f′  which are given by (4.2). The bounds on f and f′  represented by (4.1) and (4.2) respectively are sharp for the extremal function f given below by (4.5) and its derivative

Here we find the extremal points of functions belonging to the newly introduced class.

Theorem 5.1. Let f1z=z and

where θn=1−δ+nδ and =1−−1n. Then f∈TKCP(δ,k,γ,β), if and only if it takes the form

Proof. Consider

This proves that

Since

Therefore, by Defintion 1.2, we have f∈TKCP(δ,k,γ,β). Conversely, suppose that f∈TKCP(δ,k,γ,β). Then by Theorem 1.2, we have

where θn=1−δ+nδ and =1−−1n. By letting 1−∑ n=2∞δn=δ1 and

we have

Theorem 6.1. If f∈TKCP(δ,k,γ,β), then f∈Kμ for z<r0(δ,k,γ,β), we have

Proof. By a simple computation, we can write

For f∈Kμ, the class of close-to-convex of order µ, we have the condition

For the coefficient condition required by the Theorem 3.1, the inequality (3.1) is true if

or we can write

This completes the proof.

Theorem 6.2. If f∈TKCP(δ,k,γ,β), then f∈S∗μ,0≤μ<1 for  z<r1, such that

where θn=1−δ+nδ.

Proof. A simple computation leads to

For a starlike function f of order µ, we have the condition:

From the coefficient condition required by the Theorem 3.1, the inequality (6.2) is true if

where θn=1−δ+nδ. This expression yields that

where θn=1−δ+nδ. This is the desired proof.

Theorem 6.3. If f∈TKCP(δ,k,γ,β), then f∈Cμ for z<r2, where

Proof. Consider

For f∈Cμ, the class of convex functions of order µ, we have the condition:

From the condition required by the Theorem 3.1, the inequality (6.3) is true if

where θn=1−δ+nδ. This expression yields the required result.

In [19], Silverman found that the function f2(z)=z−z22 is often extremal for the family T. He applied this function to resolve his integral means inequality, conjectured in [17] and settled in [18], that is

In [18], he also proved his conjecture for the subclasses Tγ and Cγ of T. We solve Silverman's conjecture for the class of functions TKCP(δ,k,γ,β). We only need the concept of subordination between analytic functions already defined above and a subordination theorem of Littlewood [8].

Theorem 7.1. If f∈TKCP(δ,k,γ,β),0≤δ≤1,0≤β<1,k≥0 and

then for z=reiθ, and 0<r<1 we have