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

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

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.
© Kyungpook Mathematical Journal. All rights reserved.

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=2anzn,

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,         (zD,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={fA:zfHλ,zD}.

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-isin λb)|dθkπcos 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 fVkλ(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)=KConvex 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)=VkConvex 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)=Mλλ- 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)=Vkλλ- 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 fVkλ(b) if and only if

02π|Re(eiλ(1+zf(z)f(z))-(1-b)cos λ-sin λb)|dθkπcos 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 fSkλ(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)=Hλλ- 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)=Skλλ- Spirallike functions with bounded boundary rotation, [19].

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

A function fSkλ(b) if and only if

02π|Re(eiλzf(z)f(z)-(1-b)cos λ-isin λb)|dθkπcos λ.

Using Definitions 1.2 and 1.3, we obtain the following characterization

fVkλ(b)      if and only if         zfSkλ(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)=1202π1+ze-it1-ze-itdμ(t),

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

02πdμ(t)=2         and         02πdμ(t)k

for k ≥ 2.

Lemma 2.2

If pPkλ(b), then

p(z)=e-iλ(cosλ202π1+(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+eiλbcos λ(p(z)-1)=eiλp(z)-(1-b)cos λ-isin λbcos λ.

Since pPkλ(b), it follows from Lemma 2.1, we get

eiλp(z)-(1-b)cos λ-isin λbcos λ=1202π1+ze-it1-ze-itdμ(t).

Equivalently, we obtain

eiλp(z)=bcos λ202π1+ze-it1-ze-itdμ(t)+(1-b)cos λ+isinλ.

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

eiλp(z)=cos λ202π1+(2b-1)ze-it1-ze-itdμ(t)+isin 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 fλVkλ(b)if and only if there exists a functionsuch that

fλ(z)=[f(z)]be-iλcos λ,

where k ≥ 2, λ real with λ<π2and b ∈ ℂ – {0}.

Proof

Since fλVkλ(b), there exists pPkλ(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-isin λbcos λ=1202π1+ze-it1-ze-itdμ(t).

Hence

eiλ(1+zfλ(z)fλ(z))=bcos λ202π1+ze-it1-ze-itdμ(t)+(1-b)cos λ+isin 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)=1202π1+ze-it1-ze-itdμ(t).

Substituting (2.6) into (2.5), we get

eiλ(1+zfλ(z)fλ(z))=bcos λ(1+zf(z)f(z))+(1-b)cos λ+isin λ.

Calculating the above equality, we get

fλ(z)fλ(z)=be-iλcos λ(1z+f(z)f(z))+(1-b)e-iλcos λz+e-iλisin λ-1z=be-iλcos λf(z)f(z).

Integrating both sides, we obtain

ln fλ(z)=be-iλcos λln f(z).

This gives (2.4).

The following result is a consequence of Theorem 2.3.

Corollary 2.4

fλVkλ(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-iλcos λ02πlog (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

a2k2bcosλ.

This bound is sharp for the functions of the form

fλ(z)=[(1+z)k2-1(1-z)k2+1]be-iλcos 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-iλcos λ.

That is,

1+2a2z+3a3z2+=[1+2b2z+3b3z2+]be-iλcos λ.

Comparing the coefficients of z on both sides, we get

a2=b2.

In [12], Lehto proved that b2k2. Therefore we obtain

a2=b2be-iλcos λk2bcos 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-a2)

for all, thenand

|zf(z)f(z)-2z21-z2|kz1-z2.

Lemma 2.7

If fλVkλ(b), then the function Fλ defined by

Fλ(z)=fλ(z+a1+a¯z)fλ(a)(1+a¯z)2be-iλcos λ,         Fλ(0)=0

also belongs to Vkλ(b).

Proof

Let fλVkλ(b). By Theorem 2.3, there exists a function such that

fλ(z)=[f(z)]be-iλcos 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 FλVkλ(b) such that

Fλ(z)=[F(z)]be-iλcos 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-iλcos λ[f(a)]be-iλcos λ(1+a¯z)2be-iλcos λ=fλ(z+a1+a¯z)fλ(a)(1+a¯z)2be-iλcos λ,

which proves the lemma.

Theorem 2.8

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

|zfλ(z)fλ(z)-2bz2e-iλcos λ1-z2|kbzcos λ1-z2

for all.

Proof

If fλVkλ(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-a2)fλ(a)fλ(a)-2be-iλa¯cos λ.

Therefore

a2Fλ(0)2!=12{(1-a2)fλ(a)fλ(a)-2be-iλa¯cos λ}.

Replacing a by z and using Theorem 2.5, we get

|(1-z2)fλ(z)fλ(z)-2be-iλz¯cos λ|kbcos λ.

Therefore

|(1-z2)zfλ(z)fλ(z)-2be-iλz2cos λ|kbzcos 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 fλVkλ(b), k|b| cos λ < 1 and (2Reb − |b|2) cos2λ ≤ 1, then fλ maps

zr1=2kbcos λ+(kbcos λ)2-4(2Rebcos2λ-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-iλcos λ1-r2|kbrcos λ1-r2.

Thus, we get

Re(1+zfλ(z)fλ(z))1-kbrcos 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-kbrcos λ+(2Re(be-iλ)cos λ-1)r2=0.

Discriminat of this quadratic equation is

Δ=(kbcos λ)2-4(2Rebcos2λ-1)4(1-(2Reb-b2)cos2λ)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-iλcos 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 fλVkλ, then fλ is convex for

zr1=2kcos λ+(kcos λ)2-4cos 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 fλVkλ(b), then due to Theorem 2.3 we can write

fλ(z)={(δ1(z)z)k4+12(δ2(z)z)k4-12}be-iλcos 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 fλSkλ(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)=zexp [-be-iλcos λ02πlog(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=1nγjn+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 fjSkλ(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 FnVkλ(μ)for each positive integer n, where μ=bj=1nγj.

Proof

Since fj(z)=z+n=2an,jzn, we have fj(z)z0 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 λj=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-bj=1nγj)cos λ-isin λ)=j=1nγjRe(eiλzfj(z)fj(z)-(1-b)cos λ-isin λ).

On the other hand, since fjSkλ(b) for 1 ≤ jn and by using (1.5), we have

02π|Re(eiλzfj(z)fj(z)-(1-b)cos λ-isin λb)|dθkπcos λ.

On letting μ=bj=1nγj, (1.4), (4.3) and (4.4) yield

02π|Re(eiλ(1+zFn(z)Fn(z)-(1-bj=1nγj)cos λ-isin λb)|dθj=1nγj02π|Re(eiλzfj(z)fj(z)-(1-b)cos λ-isin λb)|dθkπcos λ.

Then from above inequality, we obtain

02π|Re(eiλ(1+zFn(z)Fn(z)-(1-μ)cos λ-isin λμ)|dθkπcos λ.

In view of (1.4), it follows that FnVkλ(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

01+(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 f1Skλ(b) with k ≥ 2, then the integral operator

F1(z)=0zf1(z)zdzVkλ(b).

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

Theorem 4.4

Let fjVkλ(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 GnVkλ(μ)for each positive integer n, where μ=bj=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)cosλj=1nγj on the right hand side, taking real part on both sides, and simplifying, we get

Re(eiλ(1+zGn(z)Gn(z))-(1-bj=1nγj)cos λ-isin λ)=j=1nγjRe(eiλ(1+zfj(z)fj(z))-(1-b)cos λ-isin λ).

Since fjVkλ(b) for 1 ≤ jn and by using (1.4), we obtain

02π|Re(eiλ(1+zfn(z)fj(z)-(1-b)cos λ-isin λb)|dθkπcos λ.

Therefore letting μ=bj=1nγj, (1.4), (4.5) and (4.6) give

02π|Re(eiλ(1+zGn(z)Gn(z))-(1-μ)cos λ-isin λμ)|dθkπcos λ.

It now follows from (1.4) that GnVkλ(μ) for each positive integer n.

Remark 4.5

For n = 1 and γ1 = 1, Theorem 4.4 proves that f1Vkλ(b) with k ≥ 2, then the integral operator

F1(z)=0zf1(z)dzVkλ(b).
Acknowledgements

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

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