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

Authors: Rabha W. Ibrahim
Journal: International Journal of Analysis
 (2013)

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## 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.


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## Body

## 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 orderα is defined, for a function f(z) by
(1)Dzαf(z)∶=1Γ(1-α)ddz∫0z‍f(ζ)(z-ζ)αdζ;0≤α<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 orderα is defined, for a function f(z), by
(2)Izαf(z)∶=1Γ(α)∫0z‍f(ζ)(z-ζ)α-1dζ;α>0,
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
(3)Izα,μf(z)=∫0z‍ζ1μdζ1∫0ζ1‍ζ2μdζ2⋯∫0ζn-1‍ζnμf(ζn)dζn.
Employing the Dirichlet technique implies
(4)∫0z‍ζ1μdζ1∫0ζ1‍ζμf(ζ)dζ=∫0z‍ζμf(ζ)dζ∫ζz‍ζ1μdζ1=1μ+1∫0z‍(zμ+1-ζμ+1)ζμf(ζ)dζ.
Repeating the above step n-1 times yields
(5)∫0z‍ζ1μdζ1∫0ζ1‍ζ2μdζ2⋯∫0ζn-1‍ζnμf(ζn)dζn=(μ+1)1-n(n-1)!∫0z‍(zμ+1-ζμ+1)n-1ζμf(ζ)dζ,
which imposes the fractional operator type
(6)Izα,μf(z)=(μ+1)1-αΓ(α)∫0z‍(zμ+1-ζμ+1)α-1ζμf(ζ)dζ,
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 (zμ+1-ζμ+1)-α is removed by requiring log(zμ+1-ζμ+1) to be real when (zμ+1-ζμ+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(7)Dzα,μf(z)∶=(μ+1)αΓ(1-α)ddz∫0z‍ζμf(ζ)(zμ+1-ζμ+1)αdζ;0≤α<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 (zμ+1-ζμ+1)-α is removed by requiring log(zμ+1-ζμ+1) to be real when (zμ+1-ζμ+1)>0. We have
(8)Dzα,μzν=(μ+1)α-1Γ((ν/(μ+1))+1)Γ((ν/(μ+1))+1-α)z(1-α)(μ+1)+ν-1.Let𝒜 denote the class of functions f(z) normalized by
(9)f(z)=z+∑n=2∞‍anzn,z∈U.
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 |an|≤n,n∈ℕ∖{1}. Moreover, if the function f(z) given by (9) is in the class 𝒞, then |an|≤1,n∈ℕ.In our present investigation, we will also make use of the Fox-Wright generalizationqΨp[z] of the hypergeometric qFp function defined by [12]
(10)Ψqp[(α1,A1),…,(αq,Aq);(β1,B1),…,(βp,Bp);z]=Ψqp[(αj,Aj)1,q;(βj,Bj)1,p;z]≔∑n=0∞=Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)znn!=∑n=0∞∏j=1qΓ(αj+nAj)∏j=1qΓ(βj+nBj)znn!,
where Aj>0 for all j=1,...,q,   Bj>0 for all j=1,...,p, and 1+∑j=1p‍Bj-∑j=1q‍Aj≥0 for suitable values |z|<1, and αi,βj are complex parameters.It is well known that(11)Ψqp[(α1,1),…,(αq,1);(β1,1),…,(βp,1);z]=Λ-1Fqp(α1,...,αq,β1,...,βp;z),
where
(12)Λ∶=∏j=1pΓ(βj)∏i=1qΓ(αi),
and qFp is the generalized hypergeometric function.Now by making use of the operator (7), we introduce the following extension operator Φα,μ:𝒜→𝒜:
(13)Φα,μf(z)∶=Γ((1/(μ+1))+1-α)(μ+1)α-1Γ((1/(μ+1))+1)×zα-μ+μαDzα,μf(z)=Γ((1/(μ+1))+1-α)(μ+1)α-1Γ((1/(μ+1))+1)zα-μ+μαDzα,μ×(z+∑n=2∞‍anzn)=Γ((1/(μ+1))+1-α)(μ+1)α-1Γ((1/(μ+1))+1)zα-μ+μα×[(μ+1)α-1Γ((1/(μ+1))+1)Γ((1/(μ+1))+1-α)z-α+μ-μα+1+∑n=2∞‍Γ((n/(μ+1))+1)Γ((n/(μ+1))+1-α)anzn-α+μ-μα]=z+∑n=2∞‍Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)×Γ((n/(μ+1))+1)Γ((n/(μ+1))+1-α)anzn∶=z+∑n=2∞‍ϕnα,μanzn.
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,
(14)Φα,μf(z)=z+∑n=2∞‍Γ(n+1)Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)×Γ((n/(μ+1))+1)Γ((n/(μ+1))+1-α)ann!zn=∑n=0∞‍Γ(n+1)Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)×Γ((n/(μ+1))+1)Γ((n/(μ+1))+1-α)ann!zn=Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)×Ψ21[(1,1),(1,1μ+1);(1-α,1μ+1);z]*f(z)=Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)2Ψ1[z]*f(z)∶=Ψ(α,μ;z)*f(z),
where a0=0, a1=1, and * is the Hadamard product. Note that
(15)Φ0,0f(z)=f(z),Φ1,0f(z)=zf′(z),(Φα,0f)(z)=(L(2,2-α)f)(z),
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'sG-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 functionsf 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 functionf defined by (9), the Hankel determinant of f is defined by
(16)|anan+1...an+q-1an+1an+2...an+q⋮⋮⋮⋮an+q-1an+q...an+2q-2|.
Now we proceed to define a new class of analytic function involving the operator (13).Definition 5.
The functionf∈𝒜 is said to be in the class ℛα,μ(θ,ρ), where 0≤α<1,μ≥0,|θ|<π/2,0≤ρ≤1, if it satisfies the inequality
(17)ℜ{eiθΦα,μf(z)z}>ρcosθ,(z∈U).
Consequently, from Definition 4, we have
(18)f∈ℛα,μ(θ,ρ)⟺eiθΦα,μf(z)z=[(1-ρ)p(z)+ρ]cosθ+isinθ,
where p(z)=1+c1z+c2z2+⋯,z∈U satisfies the following properties [26]:
(19)(i)|cn|≤2andℜ(p(z))>0,(ii)2c2=c12+w(4-c12),(iii)4c3=c13+2(4-c12)c1w-c1(4-c12)w2+2(4-c12)(1-w|2)z,(|w|≤1,|z|≤1).
We denote this class by 𝒫.Note that(20)ℛα,0(0,ρ)=ℜ{Φα,0f(z)z}>ρ(see [27]),(21)ℛα,0(θ,ρ)=ℜ{eiθΦα,0f(z)z}>ρcosθ(see [28]).It is well known that, for the univalent functionf 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-a32| for f∈ℛα,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.
Letϕ and ψ be univalent convex in U. Then, the Hadamard product ϕ*ψ is also univalent convex function in U.Lemma 7.
LetΦ and Ψ be univalent convex in U, and ϕ≺Φ and ψ≺Ψ. Then, ϕ*ψ≺Φ*Ψ.Lemma 8.
Letϕ and ψ be starlike of order 1/2 then, for function Φ satisfying ℜ(Φ(z))>σ(σ∈[0,1)),
(22)ℜ(ϕ(z)*Φ(z)ψ(z)ϕ(z)*ψ(z))>σ,(z∈U).


## 2. Main Results

We have the following result.Theorem 9.
Let the functionf be in the class ℛα,μ(θ,ρ). Then
(23)|a2a4-a32|≤(1-ρ)2cos2θμ12μ2μ4μ32(80β+16μ32),
where β:=|μ32-μ2μ4| and
(24)μ1=Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1),μ2=Γ((2/(μ+1))+1)Γ((2/(μ+1))+1-α),μ3=Γ((3/(μ+1))+1)Γ((3/(μ+1))+1-α),μ4=Γ((4/(μ+1))+1)Γ((4/(μ+1))+1-α).
The estimate (23) is sharp.Proof.
Sincef∈ℛα,μ(θ,ρ), then
(25)eiθΦα,μf(z)z=[(1-ρ)(1+c1z+c2z2+⋯)+ρ]cosθ+isinθ.
Comparing the coefficients of (13) and (25), we receive
(26)a2=(1-ρ)c1cosθeiθμ1μ2,a3=(1-ρ)c2cosθeiθμ1μ3,a4=(1-ρ)c3cosθeiθμ1μ4,
where
(27)μ1=Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1),μ2=Γ((2/(μ+1))+1)Γ((2/(μ+1))+1-α),μ3=Γ((3/(μ+1))+1)Γ((3/(μ+1))+1-α),μ4=Γ((4/(μ+1))+1)Γ((4/(μ+1))+1-α).
Therefore, (26) implies
(28)|a2a4-a32|=(1-ρ)2cos2θμ12|c1c3μ2μ4-c22μ32|.
By letting c1∶=c and using (i)–(iii), we have
(29)|a2a4-a32|=(1-ρ)2cos2θμ12μ2μ4μ32×|c4(μ32-μ2μ4)+2wc2(4-c2)×(μ32-μ2μ4)-w2(4-c2)×(μ32c2+μ2μ4(4-c2))+(1-|w|2)z(2c(4-c2)μ32)|.
By employing the triangle inequality and assuming |w|∶=x,β∶=|μ32-μ2μ4|,c>0, and |z|≤1, we obtain
(30)|a2a4-a32|≤(1-ρ)2cos2θμ12μ2μ4μ32×{c4β+2xc2(4-c2)β+x2(4-c2)β(c2+2c)+2c(4-c2)μ32}∶=F(x,c).
Our aim is to maximize F in the interior of the domain 𝒟=[0,1]×[0,2]. Since
(31)∂F(x,c)∂x=(1-ρ)2cos2θμ12μ2μ4μ32{2c2(4-c2)β+2x(4-c2)β(c2+2c)}>0,
thus F cannot have a maximum in the interior of 𝒟. Furthermore,
(32)maxx∈[0,1]F(x,c)=F(1,c)∶=H(c),
where
(33)H(c)≤(1-ρ)2cos2θμ12μ2μ4μ32[16β+16β(4-c2)+4(4-c2)μ32]∶=G(c).
But
(34)maxc∈[0,2]G(c)=G(0);
hence the upper bound of (28) is
(35)|a2a4-a32|≤(1-ρ)2cos2θμ12μ2μ4μ32(80β+16μ32).
The equality holds for the functions
(36)f(z)=Ψ(α,μ;z)*eiθ[z(1+(1-2ρ)z21-z2cosθ+isinθ)].Remark 10.
Lettingμ=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 thatθ∈(-π/2,π/2),ρ∈[0,1) and α1,α2∈[0,1), with α1<α2. If the subordination
(37)z𝒢′′(1μ+1+1-α2,1μ+1+1-α1;z)+𝒢′(1μ+1+1-α2,1μ+1+1-α1;z)≺1+z,
where
(38)𝒢(a,c;z):=∑n=0∞‍(a)n(c)nzn+1,(c≠0,-1,-2,...),
holds, then
(39)ℛα2,μ(θ,ρ)⊂ℛα1,μ(θ,ρ).Proof.
Letf∈ℛα2,μ(θ,ρ). We rewrite
(40)Φα1,μf(z)=Ψ(α1,μ;z)*f(z)=(Ψ(-1)(α2,μ;z)*Ψ(α2,μ;z))*Ψ(α1,μ;z)*f(z)=(Ψ(-1)(α2,μ;z)*Ψ(α1,μ;z))*Ψ(α2,μ;z)*f(z)=𝒢(1μ+1+1-α2,1μ+1+1-α1;z)*Φα2,μf(z),
where
(41)Ψ(α2,μ;z)=∑n=0∞‍Γ(n+1)Γ((1/(μ+1))+1-α)Γ((1/(μ+1))+1)×Γ((n/(μ+1))+1)Γ((n/(μ+1))+1-α)1n!zn,Ψ(-1)(α2,μ;z)=∑n=0∞‍Γ((1/(μ+1))+1)Γ(n+1)Γ((1/(μ+1))+1-α)×Γ((n/(μ+1))+1-α)Γ((n/(μ+1))+1)n!zn,𝒢(a,c;z)=∑n=0∞‍(a)n(c)nzn+1(c≠0,-1,-2,…),
and (x)k=Γ(x+k)/Γ(x) is a Pochhammer symbol. Therefore,(42)eiθΦα1,μf(z)z=𝒢((1/(μ+1))+1-α2,(1/(μ+1))+1-α1;z)*(eiθΦα2,μf(z)/z)z𝒢((1/(μ+1))+1-α2,(1/(μ+1))+1-α1;z)*z. 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
(43)ℜ(eiθΦα1,μf(z)z)>ρcosθ,
and consequently ℛα2,μ(θ,ρ)⊂ℛα1,μ(θ,ρ).Remark 12.
Condition (37) can be replaced by another condition to obtain the convexity of the function 𝒢, such that
(44)|z𝒢′′(1μ+1+1-α2,1μ+1+1-α1;z)|<12
yields that 𝒢((1/(μ+1))+1-α2,(1/(μ+1))+1-α1;z) is convex (see [31]).Theorem 13.
Letf∈𝒮* and h∈ℛα,μ(θ,ρ)(ρ∈[0,1],θ∈(-π/2,π/2),α∈[0,1),μ≥0). Then f*h∈ℛα,μ(θ,ρ).Proof.
By employing the properties of the Hadamard product, we receive(45)Φα,μ(f*h)(z)=f(z)*Φα,μh(z).
Therefore,
(46)eiθΦα,μ(f*h)(z)z=f(z)*((eiθΦα,μh(z))/z)zf(z)*z.
In virtue of Lemma 8, we have
(47)ℜ(eiθΦα,μ(f*h)(z)z)>ρcosθ.
Hence f*h∈ℛα,μ(θ,ρ).Theorem 14.
Letf∈ℛα,μ(θ,ρ)(ρ∈[0,1],θ∈(-π/2,π/2),α∈[0,1),μ≥0). Then the integral
(48)(ℐf)(z)=τ+1zτ∫0z‍ξτ-1f(ξ)dξ(z∈U,τ>-1)
is also in ℛα,μ(θ,ρ).Proof.
It is easy to show that(49)Φα,μ(ℐf)(z)=𝒢(τ+1,τ+2)*Φα,μf(z).
Therefore,
(50)eiθΦα,μ(ℐf)(z)z=𝒢(τ+1,τ+2)*((eiθΦα,μf(z))/z)z𝒢(τ+1,τ+2)*z.
But 𝒢(τ+1,τ+2)∈𝒮*(1/2), thus in view of Lemma 8, the proof is complete.Remark 15.
Whenμ=0 in Theorems 13 and 14, we have the results given in [28].Theorem 16.
Letρ∈[0,1],θ∈(-π/2,π/2),α∈[0,1),μ≥0. If the subordination
(51)𝒢′′(1μ+1+1-α,1μ+1+1;z)-2z𝒢′(1μ+1+1-α,1μ+1+1;z)+2z2𝒢(1μ+1+1-α,1μ+1+1;z)≺1+z
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.
Letf∈ℛα2,μ(θ,ρ). Then
(52)f(z)z≺g(z),
where g is defined in Theorem 16.Proof.
Sincef∈ℛα2,μ(θ,ρ), then we have
(53)Φα,μf(z)z≺e-iθ(1+(1-2ρ)z21-z2cosθ+isinθ).
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-a32|, 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.



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