Let stands the class of functions f that are analytic in the open unit disk U = {z ∈ ℂ : |z| < 1}, are normalized by the conditions f(0) = f′(0) − 1 = 0, and have the form: (1.1)f(z)=z+∑k=2∞akzk.Let S be the subclass of consisting of functions of the form (1.1) which are also univalent in U. The Koebe one-quarter theorem (see [4]) states that the image of U under every function f ∈ S contains a disk of radius 14. Therefore, every function f ∈ S has an inverse f⁻¹ which satisfies f⁻¹(f(z)) = z, (z ∈ U) and f(f⁻¹(w)) = w, (|w|<r0(f),r0(f)≥14), where (1.2)g(w)=f−1(w)=w−a2w2+(2a22−a3)w3−(5a23−5a2a3+a4)w4+⋯.A function is said to be bi-univalent in U if both f and f⁻¹ are univalent in U. We denote by Σ the class of bi-univalent functions in U satisfying (1.1). In fact, Srivastava et al. [15] has apparently revived the study of analytic and bi-univalent functions in recent years, it was followed by such works as those by Frasin and Aouf [6], Goyal and Goswami [7], Srivastava and Bansal [9] and others (see, for example [3, 10, 11, 12, 14]).

For each function f ∈ S, the function h(z)=(f(zm))1m, (z ∈ U, m ∈ ℕ) is univalent and maps the unit disk U into a region with m-fold symmetry. A function is said to be m-fold symmetric (see [8]) if it has the following normalized form: (1.3)f(z)=z+∑k=1∞amk+1zmk+1,(z∈U,m∈ℕ).We denote by S_m the class of m-fold symmetric univalent functions in U, which are normalized by the series expansion (1.3). In fact, the functions in the class S are one-fold symmetric.

In [16] Srivastava et al. defined m-fold symmetric bi-univalent functions analogues to the concept of m-fold symmetric univalent functions. They gave some important results, such as each function f ∈ Σ generates an m-fold symmetric bi-univalent function for each m ∈ ℕ. Furthermore, for the normalized form of f given by (1.3), they obtained the series expansion for f⁻¹ as follows: (1.4)g(w)=w−am+1wm+1+[(m+1)am+12−a2m+1]w2m+1−[12(m+1)(3m+2)am+13−(3m+2)am+1a2m+1+a3m+1]w3m+1+…,where f⁻¹ = g. We denote by Σ_m the class of m-fold symmetric bi-univalent functions in U. It is easily seen that for m = 1, the formula (1.4) coincides with the formula (1.2) of the class Σ. Some examples of m-fold symmetric bi-univalent functions are given as follows: (zm1−zm)1m,[12log(1+zm1−zm)]1mand[−log(1−zm)]1mwith the corresponding inverse functions (wm1+wm)1m,(e2wm−1e2wm+1)1mand(ewm−1ewm)1m,respectively.

Recently, many authors investigated bounds for various subclasses of m-fold bi-univalent functions (see [1, 2, 5, 13, 16, 17, 18]).

The purpose of the present paper is to introduce the new subclasses and of Σ_m, which involve a linear combination of the following three expressions f(z)z,  f′(z)  and  zf″(z)and find estimates on the coefficients |a_m+1| and |a_2m+1| for functions in each of these new subclasses.

In order to prove our main results, we require the following lemma.

Lemma 1.1. ([4])

If, then |c_k| ≤ 2 for each k ∈ ℕ, whereis the family of all functions h analytic in U for whichRe(h(z))>0,  (z∈U),whereh(z)=1+c1z+c2z2+⋯,  (z∈U).

Definition 2.1

A function f ∈ Σ_m given by (1.3) is said to be in the class if it satisfies the following conditions: (2.1)|arg(1+1δ[λγ(zf″(z)−2)+(γ(λ+1)+λ)f′(z)+(1−λ)(1−γ)f(z)z−1])|<απ2,and (2.2)|arg(1+1δ[λγ(wg″(w)−2)+(γ(λ+1)+λ)g′(w)+(1−λ)(1−γ)g(w)w−1])|<απ2,(z,w∈U,0≤α<1,λ≥0,0≤γ≤1,δ∈ℂ{0},m∈ℕ),where the function g = f⁻¹ is given by (1.4).

Remark 2.1

It should be remarked that the class is a generalization of well-known classes consider earlier. These classes are:

• For γ = 0, the class reduce to the class ℬ_Σm(τ, λ; α) which was introduced recently by Srivastava et al. [13];

• For γ = 0 and δ = 1, the class reduce to the class which was investigated recently by Eker [5];

• For γ = 0 and λ = δ = 1, the class reduce to the class ℋ∑,mα which was given by Srivastava et al. [16].

Remark 2.2

For one-fold symmetric bi-univalent functions, we denote the class . Special cases of this class illustrated below:

• For γ = 0 and δ = 1, the class reduce to the class ℬ_Σ(α, λ) which was investigated recently by Frasin and Aouf [6];

• For γ = 0 and λ = δ = 1, the class reduce to the class ℋ∑α which was given by Srivastava et al. [15].

Theorem 2.1

Let f ∈ (0 < α ≤ 1, λ ≥ 0, 0 ≤ γ ≤ 1, δ ∈ ℂ {0}, m ∈ ℕ) be given by (1.3). Then(2.3)|am+1|≤2α|δ||αδ(m+1)[λγ(4m(m+1)+2)+2m(λ+γ)+1]+(1−α)Ω(λ,γ,m)|and(2.4)|a2m+1|≤2α2|δ|2(m+1)Ω(λ,γ,m)+2α|δ|λγ(4m(m+1)+2)+2m(λ+γ)+1,whereΩ(λ,γ,m)=[λγ((m+1)2+1)+m(λ+γ)+1]2.

Remark 2.3

In Theorem 2.1, if we choose

• γ = 0, then we obtain the results which was proven by Srivastava et al. [13, Theorem 2.1];

• γ = 0 and δ = 1, then we obtain the results which was obtained by Eker [5, Theorem 1];

• γ = 0 and λ = δ = 1, then we obtain the results which was given by Srivastava et al. [16, Theorem 2].

For one-fold symmetric bi-univalent functions, Theorem 2.1 reduce to the following corollary:

Corollary 2.1

Let (0 < α ≤ 1, λ ≥ 0, 0 ≤ γ ≤ 1, δ ∈ ℂ {0}) be given by(1.1). Then|a2|≤2α|δ||2αδ(2γ(5λ+1)+2λ+1)+(1−α)(γ(5λ+1)+λ+1)2|and|a3|≤4α2|δ|2(γ(5λ+1)+λ+1)2+2α|δ|(2γ(5λ+1)+2λ+1),

Remark 2.4

In Corollary 2.1, if we choose

• γ = 0 and δ = 1, then we obtain the results which was proven by Frasin and Aouf [6, Theorem 2.2];

• γ = 0 and λ = δ = 1, then we obtain the results which was given by Srivastava et al. [16, Theorem 1].

Definition 3.1

A function f ∈ Σ_m given by (1.3) is said to be in the class if it satisfies the following conditions: (3.1)Re{1+1δ[λγ(zf″(z)−2)+(γ(λ−1)+λ)f′(z)+(1−λ)(1−γ)f(z)z−1]}>β,and (3.2)Re{1+1δ[λγ(wg″(w)−2)+(γ(λ+1)+λ)g′(w)+(1−λ)(1−γ)g(w)w−1]}>β,(z,w∈U,0≤β<1,λ≥0,0≤γ≤1,δ∈ℂ{0},m∈ℕ),where the function g = f⁻¹ is given by (1.4).

Remark 3.1

It should be remarked that the class is a generalization of well-known classes consider earlier. These classes are:

• For γ = 0, the class reduce to the class ℬ∑m*(τ,λ;β) which was introduced recently by Srivastava et al. [13];

• For γ = 0 and δ = 1, the class reduce to the class which was investigated recently by Eker [5];

• For γ = 0 and λ = δ = 1, the class reduce to the class ℋ_Σ,m(β) which was given by Srivastava et al. [16].

Remark 3.2

For one-fold symmetric bi-univalent functions, we denote the class . Special cases of this class illustrated below:

• For γ = 0 and δ = 1, the class reduce to the class ℬ_Σ(β, λ) which was investigated recently by Frasin and Aouf [6];

• For γ = 0 and λ = δ = 1, the class reduce to the class ℋ_Σ(β) which was given by Srivastava et al. [15].

Theorem 3.1

Let (0 ≤ β < 1, λ ≥ 0, 0 ≤ γ ≤ 1, δ ∈ ℂ {0}, m ∈ ℕ) be given by (1.3). Then(3.3)|am+1|≤2|δ|(1−β)(m+1)[λγ(4m(m+1)+2)+2m(λ+γ)+1]and(3.4)|a2m+1|≤2|δ|2(1−β)2(m+1)λγ((m+1)2+1)+m(λ+γ)+1+2|δ|(1−β)λγ(4m(m+1)+2)+2m(λ+γ)+1.

Remark 3.3

In Theorem 3.1, if we choose

• γ = 0, then we obtain the results which was proven by Srivastava et al. [13, Theorem 3.1];

• γ = 0 and δ = 1, then we obtain the results which was obtained by Eker [5, Theorem 2];

• γ = 0 and λ = δ = 1, then we obtain the results which was given by Srivastava et al. [16, Theorem 3].

For one-fold symmetric bi-univalent functions, Theorem 3.1 reduce to the following corollary:

Corollary 3.1

Let (0 ≤ β < 1, λ ≥ 0, 0 ≤ γ ≤ 1, δ ∈ ℂ {0}) be given by (1.1). Then |a2|≤2|δ|(1−β)2γ(5λ+1)+2λ+1and |a3|≤4|δ|2(1−β)2γ(5λ+1)+λ+1+2|δ|(1−β)2γ(5λ+1)+2λ+1.

Remark 3.4

In Corollary 3.1, if we choose

• γ = 0 and δ = 1, then we obtain the results which was proven by Frasin and Aouf [6, Theorem 3.2];

• γ = 0 and λ = δ = 1, then we obtain the results which was given by Srivastava et al. [15, Theorem 2].