1. Introduction
Let 𝒜(n) denote the class of functions of the form
(1)f(z)=z-∑k=n+1∞akzk (ak≥0,n∈ℕ:={1,2,3,…}),
which are analytic and univalent in the open disc
(2)Δ={z∈ℂ:|z|<1}.
A function f(z)∈𝒜(n) is star-like of complex order b, denoted as f(z)∈S*(b) if and only if it satisfies
(3)ℜ{1+1b(zf′f-1)}>0 (z∈Δ).
A function f(z)∈𝒜(n) is convex of complex order b, denoted as f(z)∈C(b) if and only if it satisfies
(4)ℜ{1+1b(zf′′f′)}>0 (z∈Δ).
These classes, S*(b), and C(b) are introduced and studied by Nasr and Aouf [1] and Wiatrowski [2].
For the two functions fj(j=1,2) given by
(5)fj(z)=z+∑k=2∞ak,jzk (j=1,2),
the Hadamard product or convolution, denoted by (f1*f2)(z), is given by
(6)(f1*f2)(z)=z+∑k=2∞ak,1ak,2zk.
Given f(z) of the form (1) and δ≥0, we define n-δ neighborhood of a function f∈𝒜(n) as
(7)Nn,δ(f)={g∈𝒜(n)∣∑k=n+1∞k|ak-bk|≤δ}.
In particular, for the identity function e(z)=z,
(8)Nn,δ(e)={g∈𝒜(n)∣∑k=n+1∞k|bk|≤δ}.
The concept of Neighborhood Nn,δ of a function f is introduced and studied by Ruscheweyh [3] and extended further by Silverman [4].
For complex numbers α1,α2,…,αq and β1,β2,…,βs (βj∈ℂ∖𝒵0-; 𝒵0-={0,-1,-2,…} for j=1,2,…,s), we define the generalized hypergeometric function Fsq(α1,α2,…,αq;β1,β2,…,βs;z) as
(9)Fsq(α1,α2,…,αq;β1,β2,…,βs;z)=∑k=0∞(α1)k(α2)k⋯(αq)kzk(β1)k(β2)k⋯(βs)kk!, (q≤s+1;=∑k=0∞(α1)k(α2)k⋯(αq)kzk(β1)k(β2)k⋯(βs)kk!5., q,s∈ℕ0:=ℕ∪{0};z∈U),
where ℕ denotes the set of all positive integers and (x)k is the Pochhammer symbol defined in terms of gamma functions as
(10)(x)k=Γ(x+k)Γ(x)={1if k=0x(x+1)⋯(x+k-1)if k∈ℕ.
Corresponding to the function gq,s(α1,β1;z) defined by
(11)gq,s(α1,β1;z)=z qFs(α1,α2,…,αq;β1,β2,…,βs;z)
recently in [5], an operator 𝒟λ,μm(α1,β1)f(z):𝒜→𝒜 is defined by
(12)𝒟λ,μ0(α1,β1)f(z)∶=f(z)*gq,s(α1,β1;z),𝒟λ,μ1(α1,β1)f(z)∶=(1-λ+μ)(f(z)*gq,s(α1,β1;z))+(λ-μ)z(f(z)*gq,s(α1,β1;z))′+λμz2(f(z)*gq,s(α1,β1;z))′′,𝒟λ,μm(α1,β1)f(z)∶=𝒟λ,μ1(𝒟λ,μm-1(α1,β1)f(z)),
where 0≤μ≤λ≤1 and m∈ℕ0. By the above definition, it is easy to note that
(13)𝒟λ,μm(α1,β1)f(z)=z+∑k=2∞[1+(k-1)(λ-μ+kμλ)]m×(α1)k-1(α2)k-1…(αq)k-1(β1)k-1(β2)k-1…(βs)k-1(k-1)!akzk.
Let us take for convenience that
(14)Bk=(α1)k-1(α2)k-1…(αq)k-1(β1)k-1(β2)k-1…(βs)k-1(k-1)!,Ck=1+(k-1)(λ-μ+kμλ).
Hence, we have
(15)𝒟λ,μm(α1,β1)f(z)=z+∑k=2∞CkmBkakzk.
For suitable values of αi′s, βj′s, q, s, m, λ, and μ, we can deduce several operators such as Sălăgean derivative operator [6], Ruscheweyh derivative operator [7], fractional calculus operator [8], Carlson-Shaffer operator [9], Dziok-Srivatsava operator [10], and also the operator introduced by Abubaker and Darus [11].
Definition 1.
For 0≤α≤1, we let A be the subclass of 𝒜(n) consisting of functions of the form (1) that satisfy
(16)|1b(11111+αz[𝒟λ,μm(α1,β1)f(z)]′)-1z[𝒟λ,μm(α1,β1)f(z)]′+αz[𝒟λ,μm(α1,β1)f(z)]′)-111111×([𝒟λ,μm(α1,β1)f(z)]′(1-α)𝒟λ,μm(α1,β1)f(z)11111+αz[𝒟λ,μm(α1,β1)f(z)]′)-1)|<γ,
where z∈Δ, b∈ℂ∖{0}, 0<γ≤1, and 𝒟λ,μm(α1,β1)f(z) are as given in (15).
Definition 2.
For 0≤α≤1 we let B be the subclass of 𝒜(n), consisting of functions of the form (1) that satisfy
(17)|1b((1-α)𝒟λ,μm(α1,β1)f(z)z 𝒟λ,μm(α1,β1)f(z)z+α[𝒟λ,μm(α1,β1)f(z)]′-1)|<γ,
where z∈Δ, b∈ℂ∖{0}, 0<γ≤1, and 𝒟λ,μm(α1,β1)f(z) are as given in (15).
By specializing the parameters involved in the above definitions, we could arrive at several known as well as new classes. For example, by taking λ=1, μ=0, q=2, s=1, α1=β1, and α2=1 and the above classes reduced to
(18)A1={f∈𝒜(n)∣|1b(1111111111111111+αz[𝔻mf(z)]′)-1z[𝔻mf(z)]′111111111111111×(111111111111+αz[𝔻mf(z)]′)-1(1-α)𝔻mf(z)11111111111111111111+αz[𝔻mf(z)]′)-1+αz[𝔻mf(z)]′)-1)|<γ},
where 𝔻mf(z) denote the Sălăgean derivative of order m given by
(19)𝔻mf(z)=z-∑k=n+1∞kmakzk,B1={f∈𝒜(n)∣|1b((1-α)𝔻mf(z)z111111111111111+α[𝒟mf(z)]′-1𝔻mf(z)z)|<γ}|1b((1-α)𝔻mf(z)z.
Similarly, on taking q=2, s=1, α1=η-1, (η>-1), α2=1, β1=1, one gets
(20)A2={f∈𝒜(n)∣|1b(+αz[Dmf(z)]′)-1z[Dmf(z)]′ ×((1-α)Dmf(z)+αz[Dmf(z)]′)-1 +αz[Dmf(z)]′)-1)|<γ},
where Dmf(z) is the operator introduced and studied by Abubaker and Darus [11] given by
(21)Dmf(z)=z-∑k=n+1∞Ckm(η+k-1k-1)zk,B2={f∈𝒜(n)∣|1b((1-α)Dmf(z)z111111111111111+α[Dmf(z)]′-1Dmf(z)z)|<γ}.
Further, by taking m=0 in the definition of the classes A and B, we could arrive at Sn(q,s,α,b,γ) and Rn(q,s,α,b,γ) which were introduced and studied by Murugusundaramoorthy et al. [12].
In this paper, we establish the coefficient inequalities for the classes A and B and several inclusion relationships involving n-δ neighborhoods of analytic univalent functions with negative and missing coefficients belonging to these classes.