Bounded Nonlinear Functional Derived by the Generalized Srivastava-Owa Fractional Differential Operator

Abstract

By making use of the generalized Srivastava-Owa fractional differential operator, a class of analytical functions is imposed. The sharp bound for the nonlinear functional associated with the Hankel determinant is computed. We consider a new technique to prove our results. Important properties such as inclusion, subordination, and Hadamard product are studied. Some recent results are included.

1. Introduction

Fractional calculus (real and complex) is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. Several different derivatives were introduced: Riemann-Liouville, Hadamard, Grunwald-Letnikov, Riesz, Erdelyi-Kober operators, and Caputo [1–7].

Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been employed for imposing, for example, the characterization properties, coefficient estimates [8], distortion inequalities [9], and convolution structures for various subclasses of analytic functions and the works in the research monographs. In [10], Srivastava and Owa defined the fractional operators (derivative and integral) in the complex 𝑧-plane ℂ as follows.

Definition 1. The fractional derivative of order 𝛼 is defined, for a function 𝑓(𝑧) by 𝐷^𝛼_𝑧𝑓(𝑧)1∶=Γ()𝑑1−𝛼𝑑𝑧^𝑧₀𝑓(𝜁)(𝑧−𝜁)^𝛼𝑑𝜁;0≤𝛼<1, (1) where the function 𝑓(𝑧) is analytical in simply-connected region of the complex 𝑧-plane ℂ containing the origin and the multiplicity of (𝑧−𝜁)^−𝛼 is removed by requiring log(𝑧−𝜁) to be real when (𝑧−𝜁)>0.

Definition 2. The fractional integral of order 𝛼 is defined, for a function 𝑓(𝑧), by 𝐼^𝛼_𝑧𝑓(𝑧)1∶=Γ(𝛼)^𝑧₀𝑓(𝜁)(𝑧−𝜁)^𝛼−1𝑑𝜁;𝛼>0, (2) where the function 𝑓(𝑧) is analytical in simply connected region of the complex 𝑧-plane (ℂ) containing the origin and the multiplicity of (𝑧−𝜁)^𝛼−1 is removed by requiring log(𝑧−𝜁) to be real when (𝑧−𝜁)>0.

In [11], the author generalized a formula for the fractional integral as follows: for natural 𝑛∈ℕ={1,2,...} and real 𝜇, the 𝑛-fold integral of the form 𝐼_𝑧^𝛼,𝜇𝑓(𝑧)=^𝑧₀𝜁^𝜇₁𝑑𝜁₁^𝜁1₀𝜁^𝜇₂𝑑𝜁₂⋯^𝜁𝑛−1₀𝜁^𝜇_𝑛𝑓𝜁_𝑛𝑑𝜁_𝑛. (3) Employing the Dirichlet technique implies ^𝑧₀𝜁^𝜇₁𝑑𝜁₁^𝜁1₀𝜁^𝜇𝑓(𝜁)𝑑𝜁=^𝑧₀𝜁^𝜇𝑓(𝜁)𝑑𝜁^𝑧_𝜁𝜁^𝜇₁𝑑𝜁₁=1𝜇+1^𝑧₀𝑧^𝜇+1−𝜁^𝜇+1𝜁^𝜇𝑓(𝜁)𝑑𝜁. (4) Repeating the above step 𝑛−1 times yields ^𝑧₀𝜁^𝜇₁𝑑𝜁₁^𝜁1₀𝜁^𝜇₂𝑑𝜁₂⋯^𝜁𝑛−1₀𝜁^𝜇_𝑛𝑓𝜁_𝑛𝑑𝜁_𝑛=𝜇+1^1−𝑛(𝑛−1)!^𝑧₀𝑧^𝜇+1−𝜁^{𝜇+1𝑛−1}𝜁^𝜇𝑓(𝜁)𝑑𝜁, (5) which imposes the fractional operator type 𝐼_𝑧^𝛼,𝜇𝑓(𝑧)=𝜇+1^1−𝛼Γ(𝛼)^𝑧₀𝑧^𝜇+1−𝜁^{𝜇+1𝛼−1}𝜁^𝜇𝑓(𝜁)𝑑𝜁, (6) where 𝛼 and 𝜇≠−1 are real numbers and the function 𝑓(𝑧) is analytic in simply connected region of the complex 𝑧-plane ℂ containing the origin and the multiplicity of (𝑧^𝜇+1−𝜁^𝜇+1)^−𝛼 is removed by requiring log(𝑧^𝜇+1−𝜁^𝜇+1) to be real when (𝑧^𝜇+1−𝜁^𝜇+1)>0. When 𝜇=0, we arrive at the standard Srivastava-Owa fractional integral. Further information can be found in [11].

Corresponding to the fractional integral operator, the fractional differential operator is 𝐷_𝑧^𝛼,𝜇𝑓(𝑧)∶=𝜇+1^𝛼Γ()𝑑1−𝛼𝑑𝑧^𝑧₀𝜁^𝜇𝑓(𝜁)𝑧^𝜇+1−𝜁^𝜇+1𝛼𝑑𝜁;0≤𝛼<1, (7) where the function 𝑓(𝑧) is analytical in simply connected region of the complex 𝑧-plane ℂ containing the origin and the multiplicity of (𝑧^𝜇+1−𝜁^𝜇+1)^−𝛼 is removed by requiring log(𝑧^𝜇+1−𝜁^𝜇+1) to be real when (𝑧^𝜇+1−𝜁^𝜇+1)>0. We have 𝐷_𝑧^𝛼,𝜇𝑧^𝜈=𝜇+1^𝛼−1Γ𝜈/𝜇+1+1Γ𝑧𝜈/𝜇+1+1−𝛼^{(1−𝛼)(𝜇+1)+𝜈−1}. (8)

Let 𝒜 denote the class of functions 𝑓(𝑧) normalized by 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛,𝑧∈𝑈. (9) Also, let 𝒮,𝒮^∗ and 𝒞 denote the subclasses of 𝒜 consisting of functions which are, respectively, univalent, starlike ℜ(𝑧𝑓′(𝑧)/𝑓(𝑧))>0, and convex ℜ(1+(𝑧²𝑓′′(𝑧)/𝑓′(𝑧)))>0 in 𝑈. It is well known that, if the function 𝑓(𝑧) given by (9) is in the class 𝒮, then |𝑎_𝑛|≤𝑛,𝑛∈ℕ⧵{1}. Moreover, if the function 𝑓(𝑧) given by (9) is in the class 𝒞, then |𝑎_𝑛|≤1,𝑛∈ℕ.

In our present investigation, we will also make use of the Fox-Wright generalization _𝑞Ψ_𝑝[𝑧] of the hypergeometric _𝑞𝐹_𝑝 function defined by [12] _𝑞Ψ_𝑝𝛼₁,𝐴₁,…,𝛼_𝑞,𝐴_𝑞;𝛽₁,𝐵₁,…,𝛽_𝑝,𝐵_𝑝;𝑧=_𝑞Ψ_𝑝𝛼_𝑗,𝐴_𝑗1,𝑞;𝛽_𝑗,𝐵_𝑗1,𝑝;𝑧∶=^∞_𝑛=0=Γ𝛼₁+𝑛𝐴₁⋯Γ𝛼_𝑞+𝑛𝐴_𝑞Γ𝛽₁+𝑛𝐵₁⋯Γ𝛽_𝑝+𝑛𝐵_𝑝𝑧^𝑛=𝑛!_∞𝑛=0∏^𝑞_𝑗=1Γ𝛼_𝑗+𝑛𝐴_𝑗∏^𝑞_𝑗=1Γ𝛽_𝑗+𝑛𝐵_𝑗𝑧^𝑛,𝑛! (10) where 𝐴_𝑗>0 for all 𝑗=1,...,𝑞,   𝐵_𝑗>0 for all 𝑗=1,...,𝑝, and 1+∑^𝑝_𝑗=1𝐵_𝑗−∑^𝑞_𝑗=1𝐴_𝑗≥0 for suitable values |𝑧|<1, and 𝛼_𝑖,𝛽_𝑗 are complex parameters.

It is well known that _𝑞Ψ_𝑝𝛼₁,1,…,𝛼_𝑞;𝛽,1₁,1,…,𝛽_𝑝;𝑧,1=Λ_𝑞⁻¹𝐹_𝑝𝛼₁,...,𝛼_𝑞,𝛽₁,...,𝛽_𝑝;𝑧, (11) where ∏Λ∶=^𝑝𝑗=1Γ𝛽_𝑗∏^𝑞_𝑖=1Γ𝛼_𝑖, (12) and _𝑞𝐹_𝑝 is the generalized hypergeometric function.

Now by making use of the operator (7), we introduce the following extension operator Φ^𝛼,𝜇∶𝒜→𝒜: Φ^𝛼,𝜇𝑓(𝑧)Γ∶=1/𝜇+1+1−𝛼𝜇+1^𝛼−1Γ1/𝜇+1+1×𝑧^{𝛼−𝜇+𝜇𝛼}𝐷_𝑧^𝛼,𝜇𝑓(𝑧)=Γ1/𝜇+1+1−𝛼𝜇+1^𝛼−1Γ𝑧1/𝜇+1+1^{𝛼−𝜇+𝜇𝛼}𝐷_𝑧^𝛼,𝜇×𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛=Γ1/𝜇+1+1−𝛼𝜇+1^𝛼−1Γ𝑧1/𝜇+1+1^{𝛼−𝜇+𝜇𝛼}×𝜇+1^𝛼−1Γ1/𝜇+1+1Γ𝑧1/𝜇+1+1−𝛼^{−𝛼+𝜇−𝜇𝛼+1}+^∞_𝑛=2Γ𝑛/𝜇+1+1Γ𝑎𝑛/𝜇+1+1−𝛼^𝑛𝑧^{𝑛−𝛼+𝜇−𝜇𝛼}=𝑧+^∞_𝑛=2Γ1/𝜇+1+1−𝛼Γ×Γ1/𝜇+1+1𝑛/𝜇+1+1Γ𝑎𝑛/𝜇+1+1−𝛼^𝑛𝑧^𝑛∶=𝑧+^∞_𝑛=2𝜙_𝑛^𝛼,𝜇𝑎_𝑛𝑧^𝑛. (13) Obviously, when 𝜇=0, we have the extension fractional differential operator defined in [13] ([14] for recent work), which contains the Carlson-Shaffer operator. In term of the Fox-Wright generalized function, Φ^𝛼,𝜇𝑓(𝑧)=𝑧+^∞_𝑛=2Γ(𝑛+1)Γ1/𝜇+1+1−𝛼Γ×Γ1/𝜇+1+1𝑛/𝜇+1+1Γ𝑎𝑛/𝜇+1+1−𝛼^𝑛𝑧𝑛!^𝑛=^∞_𝑛=0Γ(𝑛+1)Γ1/𝜇+1+1−𝛼Γ×Γ1/𝜇+1+1𝑛/𝜇+1+1Γ𝑎𝑛/𝜇+1+1−𝛼^𝑛𝑧𝑛!^𝑛=Γ1/𝜇+1+1−𝛼Γ×1/𝜇+1+1₂Ψ₁(),11,11,;1𝜇+11−𝛼,;𝑧(𝑧)=Γ𝜇+1∗𝑓1/𝜇+1+1−𝛼Γ1/𝜇+1+1²Ψ₁[𝑧](𝑧)(𝑧),∗𝑓∶=Ψ𝛼,𝜇;𝑧∗𝑓 (14) where 𝑎₀=0, 𝑎₁=1, and ∗ is the Hadamard product. Note that Φ^0,0𝑓(𝑧)=𝑓(𝑧),Φ^1,0𝑓(𝑧)=𝑧𝑓^′(𝑧),Φ^𝛼,0𝑓(𝑧)=𝐿(2,2−𝛼)𝑓(𝑧), (15) where 𝐿(𝑎,𝑐)𝑓 is the Carlson-Shaffer operator. Moreover, operator (14) can be viewed as a linear operator which is essentially analogous to the Dziok-Srivastava operator whenever used instead of the Fox-Wright generalization of the hypergeometric function.

Recently, various results, such as convolution and inclusion properties, distortion theorem, extreme points, and coefficient estimates, are proposed by many authors for the operators due to Srivastava involving the Wright function, generalized hypergeometric function, and Meijer's 𝐺-functions. These operators are Dziok-Srivastava, Srivastava-Wright, Cho-Kwon-Srivastava operator, Cho-Saigo-Srivastava operator, Jung-Kim-Srivastava, and Srivastava-Owa operators (see [15–24]). Going on in this generalization, we have finally the Erdelyi-Kober operator of fractional integration with three parameters used in [25].

Definition 3 (subordination principal). For two functions 𝑓 and 𝑔 analytical in 𝑈, we say that the function 𝑓(𝑧) is subordinated to 𝑔(𝑧) in 𝑈 and write 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝑈), if there exists a Schwartz function 𝑤(𝑧) analytical in 𝑈 with 𝑤(0)=0, and |𝑤(𝑧)|<1, such that 𝑓(𝑧)=𝑔(𝑤(𝑧)),𝑧∈𝑈. In particular, if the function 𝑔(𝑧) is univalent in 𝑈, the above subordination is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝑈)⊂𝑔(𝑈).

Definition 4. For the function 𝑓 defined by (9), the Hankel determinant of 𝑓 is defined by 𝑎_𝑛𝑎_𝑛+1...𝑎_{𝑛+𝑞−1}𝑎_𝑛+1𝑎_𝑛+2...𝑎_𝑛+𝑞𝑎⋮⋮⋮⋮_{𝑛+𝑞−1}𝑎_𝑛+𝑞...𝑎_{𝑛+2𝑞−2}. (16) Now we proceed to define a new class of analytic function involving the operator (13).

Definition 5. The function 𝑓∈𝒜 is said to be in the class ℛ_𝛼,𝜇(𝜃,𝜌), where 0≤𝛼<1,𝜇≥0,|𝜃|<𝜋/2,0≤𝜌≤1, if it satisfies the inequality ℜ𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓(𝑧)𝑧>𝜌cos𝜃,(𝑧∈𝑈). (17) Consequently, from Definition 4, we have 𝑓∈ℛ_𝛼,𝜇𝜃,𝜌⟺𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓(𝑧)𝑧=1−𝜌𝑝(𝑧)+𝜌cos𝜃+𝑖sin𝜃, (18) where 𝑝(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯,𝑧∈𝑈 satisfies the following properties [26]: (i)𝑐_𝑛𝑝(𝑧)()≤2andℜ>0,ii2𝑐₂=𝑐²₁+𝑤4−𝑐²₁,(iii)4𝑐₃=𝑐³₁+24−𝑐²₁𝑐₁𝑤−𝑐₁4−𝑐²₁𝑤²+24−𝑐²₁1−𝑤|²𝑧,(|𝑤|≤1,|𝑧|≤1). (19) We denote this class by 𝒫.

Note that ℛ_𝛼,0Φ0,𝜌=ℜ^𝛼,0𝑓(𝑧)𝑧>𝜌 (20)(see [27]),ℛ_𝛼,0𝜃,𝜌=ℜ𝑒^𝑖𝜃Φ^𝛼,0𝑓(𝑧)𝑧>𝜌cos𝜃 (21)(see [28]).

It is well known that, for the univalent function 𝑓 of the form (9), the sharp inequality |𝑎₃−𝑎²₂|≤1 holds. In the recent paper, we assume the Hankel determinant for 𝑛=2,𝑞=2 and calculate the sharp bound for the functional |𝑎₂𝑎₄−𝑎²₃| for 𝑓∈ℛ_𝛼,𝜇(𝜃,𝜌). Properties of this class are illustrated, and some well-known results are generalized. For this purpose, we need the following preliminary in the sequel, which can be found in [29].

Lemma 6. Let 𝜙 and 𝜓 be univalent convex in 𝑈. Then, the Hadamard product 𝜙∗𝜓 is also univalent convex function in 𝑈.

Lemma 7. Let Φ and Ψ be univalent convex in 𝑈, and 𝜙≺Φ and 𝜓≺Ψ. Then, 𝜙∗𝜓≺Φ∗Ψ.

Lemma 8. Let 𝜙 and 𝜓 be starlike of order 1/2 then, for function Φ satisfying ℜ(Φ(𝑧))>𝜎(𝜎∈[0,1)), ℜ𝜙(𝑧)∗Φ(𝑧)𝜓(𝑧)𝜙(𝑧)(𝑧)().∗𝜓>𝜎,𝑧∈𝑈 (22)

2. Main Results

We have the following result.

Theorem 9. Let the function 𝑓 be in the class ℛ_𝛼,𝜇(𝜃,𝜌). Then 𝑎₂𝑎₄−𝑎²₃≤1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃80𝛽+16𝜇²₃, (23) where 𝛽∶=|𝜇²₃−𝜇₂𝜇₄| and 𝜇₁=Γ1/𝜇+1+1−𝛼Γ,𝜇1/𝜇+1+1₂=Γ2/𝜇+1+1Γ,𝜇2/𝜇+1+1−𝛼₃=Γ3/𝜇+1+1Γ,𝜇3/𝜇+1+1−𝛼₄=Γ4/𝜇+1+1Γ.4/𝜇+1+1−𝛼 (24) The estimate (23) is sharp.

Proof. Since 𝑓∈ℛ_𝛼,𝜇(𝜃,𝜌), then 𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓(𝑧)𝑧=1−𝜌1+𝑐₁𝑧+𝑐₂𝑧²+⋯+𝜌cos𝜃+𝑖sin𝜃. (25) Comparing the coefficients of (13) and (25), we receive 𝑎₂=1−𝜌𝑐₁cos𝜃𝑒^𝑖𝜃𝜇₁𝜇₂,𝑎₃=1−𝜌𝑐₂cos𝜃𝑒^𝑖𝜃𝜇₁𝜇₃,𝑎₄=1−𝜌𝑐₃cos𝜃𝑒^𝑖𝜃𝜇₁𝜇₄, (26) where 𝜇₁=Γ1/𝜇+1+1−𝛼Γ,𝜇1/𝜇+1+1₂=Γ2/𝜇+1+1Γ,𝜇2/𝜇+1+1−𝛼₃=Γ3/𝜇+1+1Γ,𝜇3/𝜇+1+1−𝛼₄=Γ4/𝜇+1+1Γ.4/𝜇+1+1−𝛼 (27) Therefore, (26) implies 𝑎₂𝑎₄−𝑎²₃=1−𝜌²cos²𝜃𝜇²₁𝑐₁𝑐₃𝜇₂𝜇₄−𝑐²₂𝜇²₃. (28) By letting 𝑐₁∶=𝑐 and using (i)–(iii), we have 𝑎₂𝑎₄−𝑎²₃=1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃×𝑐⁴𝜇²₃−𝜇₂𝜇₄+2𝑤𝑐²4−𝑐²×𝜇²₃−𝜇₂𝜇₄−𝑤²4−𝑐²×𝜇²₃𝑐²+𝜇₂𝜇₄4−𝑐²+1−|𝑤|²𝑧2𝑐4−𝑐²𝜇²₃. (29) By employing the triangle inequality and assuming |𝑤|∶=𝑥,𝛽∶=|𝜇²₃−𝜇₂𝜇₄|,𝑐>0, and |𝑧|≤1, we obtain 𝑎₂𝑎₄−𝑎²₃≤1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃×𝑐⁴𝛽+2𝑥𝑐²4−𝑐²𝛽+𝑥²4−𝑐²𝛽𝑐²+2𝑐+2𝑐4−𝑐²𝜇²₃∶=𝐹(𝑥,𝑐). (30) Our aim is to maximize 𝐹 in the interior of the domain 𝒟=[0,1]×[0,2]. Since 𝜕𝐹(𝑥,𝑐)=𝜕𝑥1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃2𝑐²4−𝑐²𝛽+2𝑥4−𝑐²𝛽𝑐²+2𝑐>0, (31) thus 𝐹 cannot have a maximum in the interior of 𝒟. Furthermore, max_[]𝑥∈0,1𝐹(𝑥,𝑐)=𝐹(1,𝑐)∶=𝐻(𝑐), (32) where 𝐻(𝑐)≤1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃16𝛽+16𝛽4−𝑐²+44−𝑐²𝜇²₃∶=𝐺(𝑐). (33) But max_[]𝑐∈0,2𝐺(𝑐)=𝐺(0); (34) hence the upper bound of (28) is 𝑎₂𝑎₄−𝑎²₃≤1−𝜌²cos²𝜃𝜇²₁𝜇₂𝜇₄𝜇²₃80𝛽+16𝜇²₃. (35) The equality holds for the functions 𝑓(𝑧)=Ψ𝛼,𝜇;𝑧∗𝑒^𝑖𝜃𝑧1+1−2𝜌𝑧²1−𝑧²cos𝜃+𝑖sin𝜃. (36)

Remark 10. Letting 𝜇=0, we receive a recent result due to Mishra and Gochhayat [28]; putting 𝛼→1,𝜇=0,𝜌=0, we obtain a result given by Janteng et al. [30].

Theorem 11. Assume that 𝜃∈(−𝜋/2,𝜋/2),𝜌∈[0,1) and 𝛼₁,𝛼₂∈[0,1), with 𝛼₁<𝛼₂. If the subordination 𝑧𝒢^′′1𝜇+1+1−𝛼₂,1𝜇+1+1−𝛼₁;𝑧+𝒢^′1𝜇+1+1−𝛼₂,1𝜇+1+1−𝛼₁;𝑧≺1+𝑧, (37) where 𝒢(𝑎,𝑐;𝑧)∶=^∞_𝑛=0(𝑎)_𝑛(𝑐)_𝑛𝑧^𝑛+1,(𝑐≠0,−1,−2,...), (38) holds, then ℛ_𝛼2,𝜇𝜃,𝜌⊂ℛ_𝛼1,𝜇𝜃,𝜌. (39)

Proof. Let 𝑓∈ℛ_𝛼2,𝜇(𝜃,𝜌). We rewrite Φ^𝛼1,𝜇𝑓(𝑧)=Ψ𝛼₁(𝑧)=Ψ,𝜇;𝑧∗𝑓₍₋₁₎𝛼₂,𝜇;𝑧∗Ψ𝛼₂𝛼,𝜇;𝑧∗Ψ₁(𝑧)=Ψ,𝜇;𝑧∗𝑓₍₋₁₎𝛼₂,𝜇;𝑧∗Ψ𝛼₁𝛼,𝜇;𝑧∗Ψ₂(𝑧)1,𝜇;𝑧∗𝑓=𝒢𝜇+1+1−𝛼₂,1𝜇+1+1−𝛼₁;𝑧∗Φ^𝛼2,𝜇𝑓(𝑧), (40) where Ψ𝛼₂,𝜇;𝑧=^∞_𝑛=0Γ(𝑛+1)Γ1/𝜇+1+1−𝛼Γ×Γ1/𝜇+1+1𝑛/𝜇+1+1Γ1𝑛/𝜇+1+1−𝛼𝑧𝑛!^𝑛,Ψ⁽⁻¹⁾𝛼₂,𝜇;𝑧=^∞_𝑛=0Γ1/𝜇+1+1Γ()Γ×Γ𝑛+11/𝜇+1+1−𝛼𝑛/𝜇+1+1−𝛼Γ𝑛/𝜇+1+1𝑛!𝑧^𝑛,𝒢(𝑎,𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛(𝑐)_𝑛𝑧^𝑛+1(𝑐≠0,−1,−2,…), (41) and (𝑥)_𝑘=Γ(𝑥+𝑘)/Γ(𝑥) is a Pochhammer symbol. Therefore,𝑒^𝑖𝜃Φ^𝛼1,𝜇𝑓(𝑧)𝑧=𝒢1/𝜇+1+1−𝛼₂,1/𝜇+1+1−𝛼₁;𝑧∗𝑒^𝑖𝜃Φ^𝛼2,𝜇𝑓(𝑧)/𝑧𝑧𝒢1/𝜇+1+1−𝛼₂,1/𝜇+1+1−𝛼₁.;𝑧∗𝑧 (42) Assumption (37) implies that 𝒢((1/(𝜇+1))+1−𝛼₂,(1/(𝜇+1))+1−𝛼₁;𝑧) is convex (see [31, Theorem 1.9]) and consequently 𝒢((1/(𝜇+1))+1−𝛼₂,(1/(𝜇+1))+1−𝛼₁;𝑧)∈𝒮^∗(1/2) (Marx-Strohhäcker Theorem [32]). Moreover, the function 𝜓(𝑧)=𝑧 is starlike of order 1/2, then in view of Lemma 8, we obtain that ℜ𝑒^𝑖𝜃Φ^𝛼1,𝜇𝑓(𝑧)𝑧>𝜌cos𝜃, (43) and consequently ℛ_𝛼2,𝜇(𝜃,𝜌)⊂ℛ_𝛼1,𝜇(𝜃,𝜌).

Remark 12. Condition (37) can be replaced by another condition to obtain the convexity of the function 𝒢, such that 𝑧𝒢^′′1𝜇+1+1−𝛼₂,1𝜇+1+1−𝛼₁<1;𝑧2 (44) yields that 𝒢((1/(𝜇+1))+1−𝛼₂,(1/(𝜇+1))+1−𝛼₁;𝑧) is convex (see [31]).

Theorem 13. Let 𝑓∈𝒮^∗ and ℎ∈ℛ_𝛼,𝜇(𝜃,𝜌)(𝜌∈[0,1],𝜃∈(−𝜋/2,𝜋/2),𝛼∈[0,1),𝜇≥0). Then 𝑓∗ℎ∈ℛ_𝛼,𝜇(𝜃,𝜌).

Proof. By employing the properties of the Hadamard product, we receive Φ^𝛼,𝜇𝑓∗ℎ(𝑧)=𝑓(𝑧)∗Φ^𝛼,𝜇ℎ(𝑧). (45) Therefore, 𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓∗ℎ(𝑧)𝑧=𝑓(𝑧)∗𝑒^𝑖𝜃Φ^𝛼,𝜇ℎ(𝑧)/𝑧𝑧𝑓(𝑧).∗𝑧 (46) In virtue of Lemma 8, we have ℜ𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓∗ℎ(𝑧)𝑧>𝜌cos𝜃. (47) Hence 𝑓∗ℎ∈ℛ_𝛼,𝜇(𝜃,𝜌).

Theorem 14. Let 𝑓∈ℛ_𝛼,𝜇(𝜃,𝜌)(𝜌∈[0,1],𝜃∈(−𝜋/2,𝜋/2),𝛼∈[0,1),𝜇≥0). Then the integral ℐ𝑓(𝑧)=𝜏+1𝑧^𝜏𝑧₀𝜉^𝜏−1𝑓(𝜉)𝑑𝜉(𝑧∈𝑈,𝜏>−1) (48) is also in ℛ_𝛼,𝜇(𝜃,𝜌).

Proof. It is easy to show that Φ^𝛼,𝜇ℐ𝑓(𝑧)=𝒢(𝜏+1,𝜏+2)∗Φ^𝛼,𝜇𝑓(𝑧). (49) Therefore, 𝑒^𝑖𝜃Φ^𝛼,𝜇ℐ𝑓(𝑧)𝑧=𝒢(𝜏+1,𝜏+2)∗𝑒^𝑖𝜃Φ^𝛼,𝜇𝑓(𝑧)/𝑧𝑧𝒢().𝜏+1,𝜏+2∗𝑧 (50) But 𝒢(𝜏+1,𝜏+2)∈𝒮^∗(1/2), thus in view of Lemma 8, the proof is complete.

Remark 15. When 𝜇=0 in Theorems 13 and 14, we have the results given in [28].

Theorem 16. Let 𝜌∈[0,1],𝜃∈(−𝜋/2,𝜋/2),𝛼∈[0,1),𝜇≥0. If the subordination 𝒢^′′11𝜇+1+1−𝛼,−2𝜇+1+1;𝑧𝑧𝒢^′11𝜇+1+1−𝛼,+2𝜇+1+1;𝑧𝑧²𝒢11𝜇+1+1−𝛼,𝜇+1+1;𝑧≺1+𝑧 (51) holds, then 𝑔(𝑧)∶=(𝑒^−𝑖𝜃/𝑧)(𝒢((1/(𝜇+1))+1−𝛼, (1/(𝜇+1))+1;𝑧)∗𝑧(((1+(1−2𝜌)𝑧²)/(1−𝑧²))cos𝜃+𝑖sin𝜃)) is univalent convex function.

Proof. Condition (51) yields that 𝒢(1/(𝜇+1))+1−𝛼,(1/(𝜇+1))+1;𝑧/𝑧 is univalent convex (see [31, Theorem 1.9]) and consequently, in view of Lemma 6, 𝑔(𝑧) is univalent convex function.

Theorem 17. Let 𝑓∈ℛ_𝛼2,𝜇(𝜃,𝜌). Then 𝑓(𝑧)𝑧≺𝑔(𝑧), (52) where 𝑔 is defined in Theorem 16.

Proof. Since 𝑓∈ℛ_𝛼2,𝜇(𝜃,𝜌), then we have Φ^𝛼,𝜇𝑓(𝑧)𝑧≺𝑒^−𝑖𝜃1+1−2𝜌𝑧²1−𝑧²cos𝜃+𝑖sin𝜃. (53) But 𝑔 is univalent convex function (Theorem 16); thus by an application of Lemma 7, we obtain the desired assertion.

3. Conclusion

We defined a new fractional differential operator which generalized well-known linear and nonlinear operators such as Carlson-Shaffer operator and the Dziok-Srivastava (linear operators) and Srivastava-Owa fractional differential operators (nonlinear operator). By making uses this operator a generalized class of analytic functions is defined and studied. The sharp bound for nonlinear functional based on the second-order Hankel determinant |𝑎₂𝑎₄−𝑎²₃|, involving the generalized fractional differential operator, is computed. Several properties, depending on the Hadamard product, are imposed. We have shown that some results are generalized by recent works due to Mishra-Gochhayat, Ling-Ding, and Janteng et al. Furthermore, a new approach is introduced in the proof of Theorems 11 and 16 based on the subordination concept and employing the result due to Ponnusamy and Singh.