KYUNGPOOK Math. J. 2019; 59(2): 277-291
Spirallike and Robertson Functions of Complex Order with Bounded Boundary Rotations
Om Ahuja
Department of Mathematical Sciences, Kent State University, Ohio, 44021, U.S.A
e-mail : oahuja@kent.edu

Asena Çetinkaya*
Department of Mathematics and Computer Sciences, İstanbul Kültür University, İstanbul, Turkey
e-mail : asnfigen@hotmail.com

Yasemin Kahramaner
Department of Mathematics, İstanbul Ticaret University, İstanbul, Turkey
e-mail : ykahra@gmail.com
* Corresponding Author.
Received: February 10, 2018; Revised: April 2, 2019; Accepted: April 23, 2019; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Using the concept of bounded boundary rotation, we investigate various properties of two new generalized classes of spirallike and Robertson functions of complex order with bounded boundary rotations.

1. Introduction

Let be the unit disc {z : |z| < 1} and suppose is the class of functions analytic in satisfying the conditions f(0) = 0 and f′(0) = 1. Then each function f in has the Taylor expression

$f(z)=z+∑n=2∞anzn,$

because of the conditions f(0) = f′ (0) − 1 = 0.

Let denote the family of functions f in that map the unit disc conformally onto an image domain of bounded boundary rotation at most . The concept of functions of bounded boundary rotation was initiated by Loewner [14] in 1917. However, it was Paatero [22, 23] who systematically studied the class . In Pinchuk [25], it is proved that the functions in are close-to-convex in if 2 ≤ k ≤ 4. Brannan in [8] showed that is a subclass of the class of close-to-convex functions of order α for $α=k2-1$. For references and survey on bounded boundary rotation, one may refer to a recent survey written by Noor [20].

Failure to settle the Bieberbach conjecture for about 69 years led to the introduction and investigation of several subclasses of , the subfamily of that are univalent in the open unit disc (see [2]). In 1932, Spacek [21] proved that if f in satisfies the condition Re[(ηzf′(z))/f(z)] > 0 for all and a fixed complex number η, then f must be in . Without loss of generality, we may replace η with e, |λ| < π/2. Motivated by Spacek [21], Libera [13] in 1967 gave a geometric characterization of λ–spirallike functions f in that satisfy the condition

$Re (eiλzf′(z)f(z))>0, (z∈D,∣λ∣<π2).$

Denote the class of all such functions that satisfy (1.2) by ℋλ. We observe that , the family of all starlike functions in . In 1969, Robertson [29] introduced and studied the family

$Mλ={f∈A:zf′∈Hλ,z∈D}.$

A function f in ℳλ is called a λ–Robertson or a convex λ –spiral function. In 1991, Ahuja and Silverman [3] surveyed various subclasses of ℋλ and ℳλ, their associated properties and open problems.

Motivated by many earlier researchers [5, 6, 8, 11, 15, 16, 24, 27, 28, 29], we introduce the following:

### Definition 1.1

Let $Pkλ(b)$ be the class of functions p defined in that satisfy the property p(0) = 1 and the condition

$∫02π|Re (eiλp(z)-(1-b) cos λ-i sin λb)|dθ≤kπ cos λ,$

where k ≥ 2, λ real with $∣λ∣ <π2$, b ∈ ℂ –{0} and z = re.

When λ = 0, k = 2 and b = 1, the class $P20(1)=P$ is a well known class of functions with positive real part in . In fact, for different values of k, λ and b, $Pkλ(b)$ reduces to important subclasses studied by various researchers. For instance,

• (i) $P20(1-α)=P(α)$, (0 ≤ α < 1), Robertson [27].

• (ii) $Pk0(1)=Pk$, Pinchuk [26].

• (iv) $Pk0(1-α)=Pk(α)$, (0 ≤ α < 1), Padmanabhan [24].

• (v) $Pkλ(1)=Qkλ$, Moulis [15].

• (vi) $Pkλ(1-α)=Qkλ(α)$, (0 ≤ α < 1), Moulis [16].

### Definition 1.2

Let $Vkλ(b)$ denote the class of functions f in which satisfy the condition

$1+zf″(z)f′(z)∈Pkλ(b),$

where k, λ and b are given in Definition 1.1. If $f∈Vkλ(b)$, then f is called λ– Robertson function of complex order b with bounded boundary rotation.

We remark that the class $Vkλ(b)$ generalizes various known and unknown sub-classes of . For example, for different values of k, λ and b, we get the classes listed in the following table:

Subclasses of Name of a function in the class and References

$V20(1)=K$Convex functions
$V20(1-α)=K(α)$Convex functions of order α, 0 ≤ α < 1, [27]
$V20(b)=K(b)$Convex functions of complex order b, [32]
$Vk0(1)=Vk$Convex functions with bounded boundary rotation, [22, 23]
$Vk0(1-α)=Vk(α)$Convex functions of order α with bounded boundary rotation, 0 ≤ α < 1, [24]
$V2λ(1)=Mλ$λ- Robertson functions, [29]
$V2λ(1-α)=Vλ(α)$λ- Robertson functions of order α, 0 ≤ α < 1, [11]
$V2λ(b)=Mλ(b)$λ- Robertson functions of complex order b, [6]
$Vkλ(1)=Vkλ$λ- Robertson functions with bounded boundary rotation, [15]
$Vkλ(1-α)=Vkλ(α)$λ- Robertson functions of order α with bounded boundary rotation, 0 ≤ α < 1, [16].

In view of Definitions 1.1 and 1.2, we immediately get the following.

A function $f∈Vkλ(b)$ if and only if

$∫02π|Re(eiλ(1+zf″(z)f′(z))-(1-b) cos λ-sin λb)|dθ≤kπ cos λ.$

We next define another subclass of $Pkλ(b)$.

### Definition 1.3

Let $Skλ(b)$ denote the class of functions f in which satisfy the condition

$zf′(z)f(z)∈Pkλ(b),$

where k ≥ 2, λ real with $∣λ∣ <π2$, b ∈ ℂ – {0}. If $f∈Skλ(b)$, then f is called λ– spirallike function of complex order b with bounded boundary rotation.

For different values of k, λ and b, the class $Skλ(b)$ gives rise to several known and unknown subclasses of . For example, we obtain the following known classes:

Subclasses of Name of a function in the class and References

$S20=S*$Starlike functions
$S20(1-α)=S*(α)$Starlike functions of order α, 0 ≤ α < 1, [27]
$S20(b)=S(b)$Starlike functions of complex order b, [17]
$Sk0(1-α)=Sk(α)$Starlike functions of order α with bounded boundary rotation, 0 ≤ α < 1, [18, 24]
$S2λ(1)=Hλ$λ- Spirallike functions, [21]
$S2λ(1-α)=Hλ(α)$λ- Spirallike functions of order α, 0 ≤ α < 1, [13]
$S2λ(b)=Sλ(b)$λ- Spirallike functions of complex order b, [5]
$Skλ(1)=Skλ$λ- Spirallike functions with bounded boundary rotation, [19].

Using Definitions 1.1 and 1.3, we immediately obtain the following.

A function $f∈Skλ(b)$ if and only if

$∫02π|Re(eiλzf′(z)f(z)-(1-b) cos λ-i sin λb)|dθ≤kπ cos λ.$

Using Definitions 1.2 and 1.3, we obtain the following characterization

$f∈Vkλ(b) if and only if zf′∈Skλ(b).$

In view of the relations witnessed in 18 subclasses in the above two tables, we conclude that the notion of generalized classes $Vkλ(b)$ and $Skλ(b)$ unify several known subclasses of .

We remark that functions in $Vk0(1)$ have bounded boundary rotation. But, the functions in the class $Vkλ(1)$ with λ ≠ 0 may not have bounded boundary rotation. For properties and counter examples of the classes $Vk0(1)$ and $Vkλ(1)$, one may refer to Loewner [14] and Paatero [22, 23] .

In this paper, we investigate various properties of generalized classes $Vkλ(b)$ and $Skλ(b)$.

2. Properties of Class Vkλ(b)

The following result will be helpful in proving representation theorems for the classes $Pkλ(b)$ and $Vkλ(b)$.

### Lemma 2.1.([22])

A function f ∈ ℘k if and only if

$p(z)=12∫02π1+ze-it1-ze-itdμ(t),$

where μ is a real-valued function of bounded variation on [0, 2π] for which

$∫02πdμ(t)=2 and ∫02π∣dμ(t)∣≤k$

for k ≥ 2.

### Lemma 2.2

If $p∈Pkλ(b)$, then

$p(z)=e-iλ(cosλ2∫02π1+(2b-1)ze-it1-ze-itdμ(t)+isinλ),$

where k ≥ 2, λ real with $∣λ∣ <π2$, b ∈ ℂ – {0} and μ is real-valued function of bounded variation satisfying the conditions (2.1).

Proof

Letting

$f(z)=1+eiλb cos λ(p(z)-1)=eiλp(z)-(1-b) cos λ-i sin λb cos λ.$

Since $p∈Pkλ(b)$, it follows from Lemma 2.1, we get

$eiλp(z)-(1-b) cos λ-i sin λb cos λ=12∫02π1+ze-it1-ze-itdμ(t).$

Equivalently, we obtain

$eiλp(z)=b cos λ2∫02π1+ze-it1-ze-itdμ(t)+(1-b) cos λ+i sin λ.$

Since $∫02πdμ(t)=2$, the last equation is equivalent to

$eiλp(z)=cos λ2∫02π1+(2b-1)ze-it1-ze-itdμ(t)+i sin λ,$

where μ is a real-valued function of bounded variation on [0, 2π] and satisfies the conditions (2.1). This proves (2.2).

Motivated by several known results (see for instance [7, 16, 24]) and using Lemma 2.2, we first give the following result for the functions in the family $Vkλ(b)$.

### Theorem 2.3

A function $fλ∈Vkλ(b)$if and only if there exists a functionsuch that

$fλ′(z)=[f′(z)]be-iλ cos λ,$

where k ≥ 2, λ real with $∣λ∣ <π2$and b ∈ ℂ – {0}.

Proof

Since $fλ∈Vkλ(b)$, there exists $p∈Pkλ(b)$ such that

$1+zfλ″(z)fλ′(z)=p(z).$

By using (2.3), we can write

$eiλ(1+zfλ″(z)fλ′(z))-(1-b) cos λ-i sin λb cos λ=12∫02π1+ze-it1-ze-itdμ(t).$

Hence

$eiλ(1+zfλ″(z)fλ′(z))=b cos λ2∫02π1+ze-it1-ze-itdμ(t)+(1-b) cos λ+i sin λ.$

In view of Lemma 2.1, there exists a real-valued function μ of bounded variation on [0, 2π] satisfying conditions (2.1) such that

$1+zf″(z)f′(z)=12∫02π1+ze-it1-ze-itdμ(t).$

Substituting (2.6) into (2.5), we get

$eiλ(1+zfλ″(z)fλ′(z))=b cos λ(1+zf″(z)f′(z))+(1-b) cos λ+i sin λ.$

Calculating the above equality, we get

$fλ″(z)fλ′(z)=be-iλ cos λ(1z+f″(z)f′(z))+(1-b)e-iλ cos λz+e-iλi sin λ-1z=be-iλ cos λf″(z)f′(z).$

Integrating both sides, we obtain

$ln fλ′(z)=be-iλ cos λ ln f′(z).$

This gives (2.4).

The following result is a consequence of Theorem 2.3.

### Corollary 2.4

$fλ∈Vkλ(b)$if and only if there exists a function μ with bounded variation on [0, 2π] satisfying conditions (2.1) and

$fλ′(z)=exp [-be-iλ cos λ∫02πlog (1-ze-it) dμ(t)].$
Proof

Paatero [23] proved that if and only if there exists a function μ of bounded variation on [0, 2π] such that

$f′(z)=exp [-∫02πlog (1-ze-it) dμ(t)],$

with the conditions given in (2.1). In view of Theorem 2.3, we obtain desired result.

### Theorem 2.5

If $fλ(z)=z+a2z2+a3z3+…∈Vkλ(b)$, then

$∣a2∣ ≤k2∣b∣cosλ.$

This bound is sharp for the functions of the form

$fλ′(z)=[(1+z)k2-1(1-z)k2+1]be-iλ cos λ.$
Proof

Since $fλ(z)=z+a2z2+a3z3+…∈Vkλ(b)$, and by using Theorem 2.3 there exists a function such that

$fλ′(z)=[f′(z)]be-iλ cos λ.$

That is,

$1+2a2z+3a3z2+…=[1+2b2z+3b3z2+…]be-iλ cos λ.$

Comparing the coefficients of z on both sides, we get

$a2=b2.$

In [12], Lehto proved that $∣b2∣ ≤k2$. Therefore we obtain

$∣a2∣ = ∣b2be-iλ cos λ∣ ≤k2∣b∣cos λ.$

We need the following two lemmas to prove our next theorem.

### Lemma 2.6.([28])

Let, 2 ≤ k <and |a| < 1. If

$F(z)=f(z+a1+a¯z)-f(a)f′(a)(1-∣a∣2)$

for all, thenand

$|zf″(z)f′(z)-2∣z∣21-∣z∣2|≤k∣z∣1-∣z∣2.$

### Lemma 2.7

If $fλ∈Vkλ(b)$, then the function Fλ defined by

$Fλ′(z)=fλ′(z+a1+a¯z)fλ′(a)(1+a¯z)2be-iλ cos λ, Fλ(0)=0$

also belongs to $Vkλ(b)$.

Proof

Let $fλ∈Vkλ(b)$. By Theorem 2.3, there exists a function such that

$fλ′(z)=[f′(z)]be-iλ cos λ.$

Since , it follows from Lemma 2.6 that the function F defined by (2.9) is also in . Again, by using the converse part of Theorem 2.3, there exists a function $Fλ∈Vkλ(b)$ such that

$Fλ′(z)=[F′(z)]be-iλ cos λ.$

But, by (2.9) we have

$F′(z)=f′(z+a1+a¯z)f′(a)(1+a¯z)2,$

where |a| < 1. Therefore, we get

$Fλ′(z)=[f′(z+a1+a¯z)]be-iλ cos λ[f′(a)]be-iλ cos λ(1+a¯z)2be-iλ cos λ=fλ′(z+a1+a¯z)fλ′(a)(1+a¯z)2be-iλ cos λ,$

which proves the lemma.

### Theorem 2.8

If $fλ∈Vkλ(b)$and k|b| cos λ < 1, then fλ is univalent inand

$|zfλ″(z)fλ′(z)-2b∣z∣2e-iλ cos λ1-∣z∣2|≤k∣b∣∣z∣cos λ1-∣z∣2$

for all.

Proof

If $fλ∈Vkλ(b)$, then $Fλ′(z)$ defined by (2.11) is also in $Vkλ(b)$, by Lemma 2.7. Taking differentiation on both sides of (2.11) and letting z = 0, we get

$Fλ″(0)=(1-∣a∣2)fλ″(a)fλ′(a)-2be-iλa¯ cos λ.$

Therefore

$a2Fλ″(0)2!=12{(1-∣a∣2)fλ″(a)fλ′(a)-2be-iλa¯ cos λ}.$

Replacing a by z and using Theorem 2.5, we get

$|(1-∣z∣2)fλ″(z)fλ′(z)-2be-iλz¯ cos λ|≤k∣b∣cos λ.$

Therefore

$|(1-∣z∣2)zfλ″(z)fλ′(z)-2be-iλ∣z∣2 cos λ|≤k∣b‖z∣cos λ, ∣z∣ <1.$

Letting c = 2be cos λ and by using Ahlfor’s [1] criterion for univalence, it follows that fλ is univalent in if k|b| cos λ < 1. Dividing both sides of (2.15) by 1 − |z|2, we get (2.13).

### Remark 2.9

If f is in $Vk0(b)$, then f is univalent in whenever |b| < 1/k found by Umarani [31].

### Corollary 2.10

If $fλ∈Vkλ(b)$, k|b| cos λ < 1 and (2Reb − |b|2) cos2λ ≤ 1, then fλ maps

$∣z∣ ≤r1=2k∣b∣cos λ+(k∣b∣cos λ)2-4(2Reb cos2 λ-1).$

onto a convex domain. This result is sharp.

Proof

For |z| = r < 1, the inequality in (2.13) gives

$|(1+zfλ″(z)fλ′(z))-1-r2+2br2e-iλ cos λ1-r2|≤k∣b∣r cos λ1-r2.$

Thus, we get

$Re (1+zfλ″(z)fλ′(z))≥1-k∣b∣r cos λ+(2Re(be-iλ) cos λ-1)r21-r2.$

Right side of this inequality is positive for |z| < r1, where r1 is the positive root of the equation

$1-k∣b∣r cos λ+(2Re(be-iλ) cos λ-1)r2=0.$

Discriminat of this quadratic equation is

$Δ=(k∣b∣cos λ)2-4(2Reb cos2 λ-1)≥4(1-(2Reb-∣b∣2) cos2 λ)≥0,$

provided (2Reb−|b|2) cos2λ ≤ 1. Therefore, we obtain radius of convexity as given in (2.16). Sharp function is

$fλ′(z)=[(1+z)k2-1(1-z)k2+1]be-iλ cos λ.$

Letting b = 1 in Corollary 2.10, we obtain the following radius of convexity for the class $Vkλ$ defined in [15].

### Corollary 2.11

If $fλ∈Vkλ$, then fλ is convex for

$∣z∣ ≤r1=2k cos λ+(k cos λ)2-4 cos 2λ.$

### Remark 2.12

If we let p = 1 in Corollary 2 in Silvia [30], we observe that Silvia’s result reduces to the corresponding result given in Corollary 2.11.

3. Properties of Class Skλ(b)

For our result in this section, we need the following principal tool that was found in 1969 by Brannan [8].

### Lemma 3.1

The function f of the form (1.1), belongs toif and only if there are two functions δ1and δ2normalized and starlike insuch that

$f′(z)=(δ1(z)z)k4+12(δ2(z)z)k4-12.$

### Theorem 3.2

If a function fλ of the form (1.1) belongs to $Vkλ(b)$, then there exist two normalized λ−spirallike functions T1, T2insuch that

$fλ′(z)={(T1(z)z)k4+12(T2(z)z)k4-12}b.$
Proof

In view of Lemma 3.1, if and only if

$f′(z)=(δ1(z)z)k4+12(δ2(z)z)k4-12,$

where δ1, δ2 are normalized starlike functions in . If $fλ∈Vkλ(b)$, then due to Theorem 2.3 we can write

$fλ′(z)={(δ1(z)z)k4+12(δ2(z)z)k4-12}be-iλ cos λ.$

It is well-known that if δ is a starlike function in , then T(z) = z[δ(z)/z]e cos λ is λ– spirallike function (see [7]). Now using this representation, we get desired result follows from (3.2).

### Remark 3.3

If we let b = 1 − α, there exist λ–spirallike functions T1, T2 in such that

$fλ′(z)={(T1(z)z)k4+12(T2(z)z)k4-12}1-α,$

which was obtained by Moulis [16].

Using (1.6) and Corollary 2.4, we also obtain the following representation theorem for $Skλ(b)$.

### Corollary 3.4

A function $fλ∈Skλ(b)$if and only if there exists a function μ with bounded variation on [0, 2π] satisfying conditions in (2.1) such that

$fλ(z)=z exp [-be-iλ cos λ∫02πlog (1-ze-it) dμ(t)].$
4. Integral Operators

In [9], Breaz and Breaz; in [10], Breaz et al. studied the following integral operators

$Fn(z)=∫0z(f1(t)t)γ1…(fn(t)t)γndt,$$Gn(z)=∫0z(f1′(t))γ1…(fn′(t))γndt,$

for γj > 0, 1 ≤ jn and $∑j=1nγj≤n+1$. They studied starlikeness and convexity of the operators (4.1) and (4.2). In this section, we investigate these integral operators (4.1) and (4.2) for the classes $Skλ(b)$ and $Vkλ(b)$.

### Theorem 4.1

Let $fj∈Skλ(b)$for 1 ≤ jn with k ≥ 2, b ∈ ℂ – {0}. Also, let λ be a real number with $∣λ∣ <π2$, γj > 0 (1 ≤ jn). Then $Fn∈Vkλ(μ)$for each positive integer n, where $μ=b∑j=1nγj$.

Proof

Since $fj(z)=z+∑n=2∞an,jzn$, we have $fj(z)z≠0$ for all . Therefore by (4.1), we have

$Fn″(z)Fn′(z)=∑j=1nγj(fj′(z)fj(z)-1z),$

or equivalently

$eiλ(1+zFn″(z)Fn′(z))=eiλ+∑j=1nγjeiλzfj′(z)fj(z)-∑j=1nγjeiλ=eiλ+(1-b) cos λ∑j=1nγj+∑j=1nγj(eiλzfj′(z)fj(z)-(1-b) cos λ)-∑j=1nγjeiλ.$

Taking real part on both sides, we get

$Re(eiλ(1+zFn″(z)Fn′(z))-(1-b∑j=1nγj) cos λ-i sin λ)=∑j=1nγjRe(eiλzfj′(z)fj(z)-(1-b) cos λ-i sin λ).$

On the other hand, since $fj∈Skλ(b)$ for 1 ≤ jn and by using (1.5), we have

$∫02π|Re(eiλzfj′(z)fj(z)-(1-b) cos λ-i sin λb)|dθ≤kπ cos λ.$

On letting $μ=b∑j=1nγj$, (1.4), (4.3) and (4.4) yield

$∫02π|Re(eiλ(1+zFn″(z)Fn′(z)-(1-b∑j=1nγj) cos λ-i sin λb)|dθ≤∑j=1nγj∫02π|Re(eiλzfj′(z)fj(z)-(1-b) cos λ-i sin λb)|dθ≤kπ cos λ.$

Then from above inequality, we obtain

$∫02π|Re(eiλ(1+zFn″(z)Fn′(z)-(1-μ) cos λ-i sin λμ)|dθ≤kπ cos λ.$

In view of (1.4), it follows that $Fn∈Vkλ(μ)$ for each positive integer n.

We deduce the following known result from Theorem 4.1. We observed in Section 1 that $Sk0(1-α)=Sk(α)$ and $Vk0(1-α)=Vk(α)$ for 0 ≤ α < 1 and k ≥ 2. By letting λ = 0 and b = 1 − α, Theorem 4.1 gives the following result obtained in [18].

### Corollary 4.2

Letfor 1 ≤ jn with 0 ≤ α < 1 and k ≥ 2. Also, let γj > 0 (1 ≤ jn).If

$0≤1+(α-1)∑j=1nγj<1,$

thenwith $β=1+(α-1)∑j=1nγj$.

### Remark 4.3

For n = 1 and γ1 = 1, Theorem 4.1 proves that if $f1∈Skλ(b)$ with k ≥ 2, then the integral operator

$F1(z)=∫0zf1(z)zdz∈Vkλ(b).$

In particular, for n = 1, γ1 = 1, λ = 0, b = 1 and k = 2, Theorem 4.1 proves that if , then $∫0zf1(z)zdz∈K$. This is the famous result found by Alexander [4].

### Theorem 4.4

Let $fj∈Vkλ(b)$for 1 ≤ jn with k ≥ 2, b ∈ ℂ – {0}. Also, let λ be a real number with $∣λ∣ <π2$, γj > 0 (1 ≤ jn). Then the integral operator $Gn∈Vkλ(μ)$for each positive integer n, where $μ=b∑j=1nγj$.

Proof

In view of (4.2), we have

$Gn″(z)Gn′(z)=γ1f1″(z)f1′(z)+…+γnfn″(z)fn′(z),$

or equivalently

$eiλ(1+zGn″(z)Gn′(z))=eiλ+∑j=1nγjeiλ(1+zfj″(z)fj′(z))-∑j=1nγjeiλ.$

Subtracting and adding $(1-b)cosλ∑j=1nγj$ on the right hand side, taking real part on both sides, and simplifying, we get

$Re (eiλ(1+zGn″(z)Gn′(z))-(1-b∑j=1nγj) cos λ-i sin λ)=∑j=1nγjRe (eiλ(1+zfj″(z)fj′(z))-(1-b) cos λ-i sin λ).$

Since $fj∈Vkλ(b)$ for 1 ≤ jn and by using (1.4), we obtain

$∫02π|Re(eiλ(1+zfn″(z)fj′(z)-(1-b) cos λ-i sin λb)|dθ≤kπ cos λ.$

Therefore letting $μ=b∑j=1nγj$, (1.4), (4.5) and (4.6) give

$∫02π|Re(eiλ(1+zGn″(z)Gn′(z))-(1-μ) cos λ-i sin λμ)|dθ≤kπ cos λ.$

It now follows from (1.4) that $Gn∈Vkλ(μ)$ for each positive integer n.

### Remark 4.5

For n = 1 and γ1 = 1, Theorem 4.4 proves that $f1∈Vkλ(b)$ with k ≥ 2, then the integral operator

$F1(z)=∫0zf1′(z)dz∈Vkλ(b).$
Acknowledgements

We thank the referee for his/her insightful suggestions and scholarly guidance to improve the results in the present form.

References
1. LV. Ahlfors. Sufficient conditions for quasiconformal extension. Ann Math Studies, 79, Princeton Univ. Press, Princeton, N.J, 1974.
2. OP. Ahuja. The Bieberbach conjecture and its impact on the developments in geometric function theory. Math Chronicle., 15(1986), 1-28.
3. OP. Ahuja, and H. Silverman. A survey on spiral-like and related function classes. Math Chronicle., 20(1991), 39-66.
4. JW. Alexander. Functions which map the interior of the unit circle upon simple regions. Ann Math., 17(1915), 12-22.
5. FM. Al-Oboudi, and MM. Haidan. Spirallike functions of complex order. J Natural Geom., 19(2000), 53-72.
6. MK. Aouf, FM. Al-Oboudi, and MM. Haidan. On some results for λ–Spirallike and λ –Robertson functions of complex order. Publ De L’Institut Math., 77(91)(2005), 93-98.
7. T. Basgöze, and FR. Keogh. The Hardy class of a spiral-like function and its derivative. Proc Amer Math Soc., 26(1970), 266-269.
8. DA. Brannan. On functions of bounded boundary rotation I. Proc Edinburg Math Soc., 16(1968/69), 339-347.
9. D. Breaz, and N. Breaz. Two integral operators. Studia Univ Babeş-Bolyai Math., 47(2002), 13-19.
10. D. Breaz, S. Owa, and N. Breaz. A new integral univalent operator. Acta Univ Apulensis Math Inform., 16(2008), 11-16.
11. PN. Chichra. Regular functions f(z) for which zf′(z) is α–Spiral-like. Proc Amer Math Soc., 49(1975), 151-160.
12. O. Lehto. On the distortion of conformal mappings with bounded boundary rotation. Ann Acad Sci Fennicae Ser AI Math-Phys., 124(1952), 14 pp.
13. RJ. Libera. Univalent α–Spiral functions. Canad J Math., 19(1967), 449-456.
14. C. Loewner. Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises |z| < 1, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden. Ber Verh Sächs Gess Wiss Leipzig., 69(1917), 89-106.
15. EJ. Moulis. A generalization of univalent functions with bounded boundary rotation. Trans Amer Math Soc., 174(1972), 369-381.
16. EJ. Moulis. Generalizations of the Robertson functions. Pacific J Math., 81(1979), 167-174.
17. MA. Nasr, and MK. Aouf. Starlike function of complex order. J Natural Sci Math., 25(1985), 1-12.
18. KI. Noor, M. Arif, and W. Ul-Haq. Some properties of certain integral operators. Acta Univ Apulensis Math Inform., 21(2010), 89-95.
19. KI. Noor, N. Khan, and MA. Noor. On generalized Spiral-like analytic functions. Filomat., 28(7)(2014), 1493-1503.
20. KI. Noor, B. Malik, and S. Mustafa. A survey on functions of bounded boundary and bounded radius rotation. Appl Math E-Notes., 12(2012), 136-152.
21. L. Spacek. Contribution à la théorie des fonctions univalentes. Časopis Pěst Mat., 62(1932), 12-19.
22. V. Paatero. Über die konforme Abbildung von Gebieten deren Ränder von beschränkter Drehung sind. Ann Acad Sci Fenn Ser A., 33(1931), 1-77.
23. V. Paatero. Über Gebiete von beschränkter Randdrehung. Ann Acad Sci Fenn Ser A., 37(1933), 1-20.
24. KS. Padmanabhan, and R. Parvatham. Properties of a class of functions with bounded boundary rotation. Ann Polon Math., 31(3)(1975/76), 311-323.
25. B. Pinchuk. A variational method for functions of bounded boundary rotation. Trans Amer Math Soc., 138(1969), 107-113.
26. B. Pinchuk. Functions of bounded boundary rotation. Israel J Math., 10(1971), 6-16.
27. MS. Robertson. On the theory of univalent functions. Ann of Math., 37(1936), 374-408.
28. MS. Robertson. Coefficients of functions with bounded boundary rotation. Canad J Math., 21(1969), 1477-1482.
29. MS. Robertson. Univalent functions f(z) for which zf′(z) is spirallike. Michigan Math J., 16(1969), 97-101.
30. EM. Silvia. A note on special classes of p–valent functions. Rocky Mountain J Math., 9(2)(1979), 365-370.
31. PG. Umarani. Functions of bounded boundary rotation of complex order. Math Balkanica., 3(1989), 34-43.
32. P. Wiatrowski. The coefficients of a certain family of holomorphic functions. Zeszyty Nauk Univ Lodz Nauki Math Przyrod Ser II., 39(1971), 75-85.