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

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 kπ. 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^iλ, |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

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

A function f in ℳ^λ is called a λ–Robertson or a convex λ –spiral function. In 1991, Ahuja and Silverman [3] surveyed various subclasses of ℋ^λ and ℳ^λ, 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

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

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

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

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

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

Using Definitions 1.2 and 1.3, we obtain the following characterization

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).

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

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

for k ≥ 2.

Lemma 2.2

If p∈Pkλ(b), then

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).

Theorem 2.3

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

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

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

Theorem 2.5

If fλ(z)=z+a2z2+a3z3+…∈Vkλ(b), then

This bound is sharp for the functions of the form

Lemma 2.6.([28])

Let, 2 ≤ k < ∞ and |a| < 1. If

for all, thenand

Lemma 2.7

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

also belongs to Vkλ(b).

Theorem 2.8

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

for all.

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 f_λ maps

onto a convex domain. This result is sharp.

Corollary 2.11

If fλ∈Vkλ, then f_λ is convex for

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.

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

Theorem 3.2

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

Remark 3.3

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

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

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

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

Corollary 4.2

Letfor 1 ≤ j ≤ n with 0 ≤ α < 1 and k ≥ 2. Also, let γ_j > 0 (1 ≤ j ≤ n).If

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

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

Remark 4.5

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

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