# Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk

Authors: Rabha W. Ibrahim; Rafida M. Elobaid; Suzan J. Obaiys
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetricq-differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.


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## Body

## 1. Introduction

In this effort, we deal with the structure ofq-calculus, which develops an interesting technique for calculations and organizes different classes of operators and specific transformations. The significance of q-calculus appeared in a huge number of applications including physical problems. The symmetric q-activation normally achieves q-difference equations (may involve derivative). A close connection between these operators and symmetries of q-symmetric operator is accordingly to be estimated (see [1–9]). In recent investigation, we deliver a process for deriving and interpreting from a symmetry possessions and infirm analogy with the traditional cases. By combining the q-calculus and the symmetric Salagean differential operator, we introduce a novel modified symmetric Salagean q-differential operator. Via employing this operator, we deliver new classes of analytic functions.


## 2. Preliminaries

This section gives out the mathematical processing to deliver the suggested SDOs and complex conformable operator for some classes of analytic functions in the open unit disk∪=ξ∈ℂ:∣ξ∣<1. Let ∧ be the category of smooth function elicited as pursue
(1)γξ=ξ+∑n=2∞γnξn,ξ∈∪.A functionγ∈∧ is known as a starlike with respect to ξ=0 if the straight line segment combining the origin to all else point of γ embedding completely in γξ:∣ξ∣<1. The aim is that each point of γξ:∣ξ∣<1 must be manifested via (0,0). A univalent function (γ∈Ⓢ;) is indicated to be convex in ∪ if the linear slice combining two ends of γξ:∣ξ∣<1 stays completely in γξ:∣ξ∣<1. We denote these classes by S∗ and C for starlike and convex, respectively. In addition, suppose that the category P involves all functions γ analytic in ∪ with a positive real part in ∪ achieving γ0=1. Mathematically, γ∈S∗ if and only if ξγ′ξ/γξ∈P, and γ∈C if and only if 1+ξγ′′ξ/γ′ξ∈P; equivalently,
(2)Rξγ′ξ/γξ>0,for starlikeness and
(3)1+Rξγ′′ξ/γ′ξ>0,for convexity.For two functions,γ1 and γ2 belong to the category ∧ and are said to be subordinate, noting by γ1≺γ2, if we can find a Schwarz function ⊺ with ⊺0=0 and ∣⊺ξ∣<1 achieving γ1ξ=γ2⊺ξ,ξ∈∪ (the detail can be located in [10]). Obviously, γ1ξ≺γ2ξ implies that γ10=γ20 and γ1∪⊂γ2∪. We employ next facts, one can find it in [11].For every nonnegative integern, the q-integer number n symbolized by n,q is structured by
(4)n,q=1−qn1−q,where 0,q=0, 1,q=1, and limq→1−n,q=n. Consequently, the analytic function γ is written by the formula
(5)Δqγξ=γqξ−γξqξ−ξ,ξ∈∪.Clearly, we haveΔqξn=n,qξn−1. Consequently, for γ∈∧, we attain
(6)Δqγξ=∑n=1∞γnn,qξn−1,ξ∈∪,γ1=1.Forγ∈∧, it realized that the Sàlàgean q-differential operator [12] has the formula
(7)Sq0γξ=γξ,Sq1γξ=ξΔqγξ,⋯,Sqkγξ=ξΔqSqk−1γξ,such that k represents as a positive integer. A computation based on the definition of Δq implies that Sqkγξ=γξ∗Πqkξ, where ∗ is the convolution product.
(8)Πqkξ=ξ+∑n=2∞n,qkξn,and
(9)Sqkγξ=ξ+∑n=2∞n,qkγnξn.Clearly,(10)limq→1−Sqkγξ=ξ+∑n=2∞nkγnξn,the well-known Salagean differential operator [6].Consider the roleγξ and a constant 0≤λ≤1, we introduce the q-symmetric Salagean differential operator using the definition of Δq as follows:
(11)Sqλ,0γξ=γξ,Sqλ,1γξ=λξΔqγξ−1−λξΔqγ−ξ=ξλΔqγξ−1−λΔqγ−ξ=λξ+∑n=2∞n,qγnξn−1−λ−ξ+∑n=2∞n,q−1nγnξn=ξ+∑n=2∞n,qλ−1−λ−1nγnξn,Sqλ,2γξ=Sqλ,1Sqλ,1γξ=ξ+∑n=2∞n,q2λ−1−λ−1n2γnξn,⋮Sqλ,kγξ=Sqλ,1Sqλ,k−1γξ=ξ+∑n=2∞n,qkλ−1−λ−1nkγnξn.Obviously, we indicate that whenλ=1Sq1,k=Sqk, we get the Salagean q-differential operator. We can call (11) as the symmetric Sàlàgean q-differential operator in ∪. Also, we have the following two limits.
(12)limq→1−Sq1,kγξ=ξ+∑n=2∞nkγnξn,which are represented to the well-known Salagean differential operator [6] and
(13)limq→1−Sqλ,kγξ=ξ+∑n=2∞nkλ−1−λ−1nkγnξn,the symmetric differential operator [8], respectively.Depending on the definition of (11), we impose the recognizing classes:Definition 1.
A memberγ∈∧ is called in the category Sq∗λ,k,ℏ when
(14)Sq∗λ,k,ℏ=γ∈∧ξSqλ,kγξ′′Sqλ,kγξ≺ℏξ,ℏ∈C.
We obtain the following special cases:(i)
limq→1−Sq∗λ,k,ℏ=S1∗λ,k,ℏ ([8])(ii)
Sq∗λ,0,1+Aξ/1+Bξ ([13–15])(iii)
Sq∗λ,0,2/1+e−ξ ([16])(iv)
Sq∗λ,0,1+ε2ξ2/1−ε2−ε2ξ2,ε=1−5/2 ([17, 18])(v)
Sq∗λ,0,1+β−α/πilog1−exp2πi1−α/β−αξ/1−ξ ([19])(vi)
Sq∗λ,0,1+2/π1−αilog1−expπi1−α2ξ/1−ξ ([20])(vii)
Sq∗λ,0,1+ξ ([21])(viii)
Sq∗λ,0,sinξ+1 ([22])(ix)
Sq∗λ,0,cosξ+1 ([23])(x)
Sq∗λ,0,1+ξ/1+1−v/vξ1/ν,v≥0.5,ν≥1 ([24])Definition 2.
Ifγ∈∧, then γ∈Jqλ,♭A,B,k if and only if
(15)1+1♭2Sqλ,k+1γξSqλ,kËξ−Sqλ,kγ−ξ≺1+Aξ1+Bξξ∈∪,−1≤B<A≤1,k∈ℤ+,♭∈ℂ0,λ∈0,1.(i)
λ=1,q⟶1−⇒ [25](ii)
λ=1,B=0,q⟶1−⇒ [26](iii)
λ=1,A=1,B=−1,#=2,q⟶1−⇒ [27](iv)
q⟶1−⇒ [8]
In our investigation, we focus on the geometric presentation of the special classesSq∗λ,k,ℏ and Jqλ,♭A,B,k via utilizing the basis information given in [11].Lemma 3.
Leta∈ℂ, let n be a positive integer and let
(16)Ha,n=γ:γξ=a+ξ+γnξn+γn+1ξn+1+⋯.(i)
Suppose that℘∈ℝ such that Rγξ+℘ξγ′ξ>0⇒Rγξ>0. In addition, let ℘>0 and γ∈H1,n; thereafter, there are two fixed positive numbers ℓ>0 and ♭>0 with ♭=♭℘,ℓ,n so that
(17)γξ+℘ξγ′ξ≺1+ξ1−ξ♭⇒γξ≺1+ξ1−ξℓ.(ii)
Assume thatν∈0,1 and v∈H1,n, subsequently, there occurs a fixed number ℓ>0 with ℓ such that
(18)Rγ2ξ+2γξ.ξγ′ξ>ν⇒Rγξ>ℓ.(iii)
Considerγ∈Ha,n with Ra>0 posterity
(19)Rγξ+ξγ′ξ+ξ2γ′′ξ>0,or for θ:U⟶ℝ such that
(20)Rγξ+θξξγ′ξγξ>0,thus Rγξ>0.Remark 4.
Concerning Lemma3 (i), more information about ℓ and ♭ can be found in [11] (Theorem 3.1c, p. 73). About (ii), more information about ν and ℓ can be found in [11] (Theorem 3.1e or Corollary 3.1e.2, p. 77 and p. 79, respectively). And for (iii), one can see Theorem 4.1g in [11] (p. 198) and, for θξ=1, the inequality contains a Briot-Bouquet formula.


## 3. Main Results

Here, we focus in the geometric representations of the classesSq∗λ,k,ℏ and Jqλ,♭A,B,k and the outcomes of these classes.Theorem 5.
Forγ∈∧ if one of the recognizing determinations are indicated by
(i)
Sqλ,kγξ is of bounded turning(ii)
γ satisfies the subordination formula
(21)Sqλ,kγξ′≺1+ξ1−ξ♭,♭>0,ξ∈∪;(iii)
γ achieves the formula
(22)RSqλ,kγξ′Sqλ,kγξξ>ς2,ς∈0,1,ξ∈∪,(iv)
ψ satisfies the relation
(23)RξSqλ,kγξ′′−Sqλ,kγξ′+2Sqλ,kγξξ>0,(v)
γ admits the inequality
(24)RzSqλ,kγξ′Sqλ,kγξ+2Sqλ,kγξξ>1,then Sqλ,kγξ/ξ∈Pσ where 0≤σ<1.Proof.
Sort out a functionρ in the following construction:
(25)ρξ=Sqλ,kγξξ⇒ξγ′ξ+ρξ=Sqλ,kγξ′.
Via the main information,Sqλ,kγξ is of constrained limit turning; it infers that Rξρ′ξ+ρξ>0. Accordingly, via Lemma 3 (i), we acquire Rρξ>0 which incomes the first requested statement of the theorem. In opinion of the additional information, we obligate the subsequent subordination relative
(26)Sqλ,kγξ′=ξρ′ξ+ρξ≺1+ξ1−ξ♭.
Now, based on Lemma3 (i), there exists a fixed positive number ℓ>0 satisfying ♭=♭ℓ and the subordination inequality
(27)Sqλ,kγξξ≺1+ξ1−ξℓ.
This leads to the conclusion(28)RSqλ,kγξξ>σ,σ∈0,1.
Lastly, we assume the third fact, a direct reckoning reaches to(29)Rρ2ξ+2ρξ.ξρ′ξ=2RSqλ,kγξ′Sqλ,kγξξ≥ς.
According to the virtue of Lemma3 (ii), there occurs a fixed positive number, say ℓ>0 achieving Rρξ>ℓ. Consequently, we obtain
(30)ρξ=Sqλ,kγξξ∈Pσ,σ∈0,1.
Hence, via Equation (29), it indicates that
(31)RSqλ,kγξ′>0.
So by Noshiro-Warschawski and Kaplan Propositions,Sqλ,kγξ∈Ⓢ and of bounded turning in ∪.
Via the derivative (25) and operating the real, one can attain the real relation
(32)Rρξ+ξρ′ξ+ξ2ρ′′ξ=RξSqλ,kγξ′′−Sqλ,kγξ′+2Sqλ,kγξξ>0.
Hence, according to Lemma3 (ii), one get RSqλ,kγξ/ξ>0.
Via considering the logarithmic derivative on (25) and acting as areal part, we obtain the consequence conversation:
(33)Rρξ+ξρ′ξρξ+ξ2ρ′′ξ=RξSqλ,kγξ′′Sqλ,kγξ+2Sqλ,kγξξ−1>0.
Thus, according to Lemma3 (iii), where θξ=1, we attain that RSqλ,kγξ/ξ>0.Theorem 6.
Suppose thatγ∈Sq∗λ,k,ℏ, with ℏξ∈C. Then
(34)Sqλ,k≺ξe∫0ξℏðu−1/udu,where ð∈∪ (analytic) satisfies that ð0=0 and ∣ðξ∣<1. In addition, for ∣ξ∣=χ, Sqλ,k reckoning fulfills the formula statement
(35)exp∫01ℏð−χ−1χdχ≤Sqλ,kξ≤exp∫01ℏðχ−1χdχ.Proof.
We note thatγ∈Sq∗λ,k,ℏ, then one can gain
(36)zSqλ,kγξ′Sqλ,kγξ≺ℏξ,ξ∈∪,this confirms that there occurs analytic function type Schwarz achieving the relations ð0=0,∣ðξ∣<1 and
(37)ξSqλ,kγξ′Sqλ,kγξ=ℏðξ,ξ∈∪.
A calculation gives us(38)Sqλ,kγξ′Sqλ,kγξ−1ξ=ℏðξ−1ξ.
Via integrating left and right parts, one can achieve(39)logSqλ,kγξ−logξ=∫0ξℏðu−1udu.
Thus, we have(40)logSqλ,kγξξ=∫0ξℏðu−1udu.
Via the definition and the properties of subordination, one can have(41)Sqλ,kγξ≺ξexp∫0ξℏðu−1udu.
Furthermore, we deliver thatℏz maps the disk 0<∣ξ∣<χ<1 onto a convex symmetric domain corresponding to the real axis, that is
(42)ℏ−χ∣ξ∣≤Rℏðχξ≤ℏχ∣ξ∣,χ∈0,1,which implies that
(43)ℏ−χ≤ℏ−χξ,ℏχξ≤ℏχ,∫01ℏð−χ∣ξ∣−1χdχ≤R∫01ℏðχ−1χdχ≤∫01ℏðχ∣ξ∣−1χdχ.
Via applying Equation (40), one can indicate that
(44)∫01hð−χ∣ξ∣−1χdχ≤log∣Sqλ,kγξξ∣≤∫01ℏðχ∣ξ∣−1χdχ,which leads to
(45)exp∫01hð−χ∣ξ∣−1χdχ≤∣Sqλ,kγξξ∣≤exp∫01ℏðχ∣ξ∣−1χdχ.
Hence, we have(46)exp∫01hð−χ−1ηdχ≤∣Sqλ,kγξξ∣≤exp∫01ℏðχ−1χdχ.Corollary 7 [8].
Ifq⟶1 in Theorem 6, then
(47)S1λ,kγξ≺ξexp∫0ξℏðu−1udu.Theorem 8.
Suppose thatγ∈Jqλ,♭A,B,k, then the odd construction formula
(48)Mξ=12γξ−γ−ξ,ξ∈∪,fulfills the consequently subordination
(49)1+1♭Sqλ,k+1MξSqλ,kMξ−1≺1+Aξ1+Bξ,RξMξ′Mξ≥1−ϱ21+ϱ2,∣ξ∣=ϱ<1,−1≤B<A≤1,ξ∈∪,k=1,2,⋯,λ∈0,1,♭∈ℂ\0.Proof.
Letγ∈Jqλ,♭A,B,k. Subsequently, we get that there occurs a function L∈JA,B with the formula
(50)Lξ−1.♭=2Sqλ,k+1γξSqλ,kγξ−Sqλ,kγ−ξ,L−ξ−1.♭=−2Sqλ,k+1γ−ξSqλ,kγξ−Sqλ,kγ−ξ.
This yields(51)1+1♭Sqλ,k+1MξSqλ,kMξ−1=Lξ+L−ξ2.
In addition, sinceL achieves the inequality
(52)Lξ≺1+Aξ1+Bξ,taking account that the fractional functional express 1+Aξ/1+Bξ is univalent and hence, consequently, we attain the relation
(53)1+1♭Sqλ,k+1MξSqλ,kMξ−1≺1+Aξ1+Bξ.
Furthermore, the expressionMξ∈S∗, which leads to the inequality
(54)ξMξ′Mξ≺1−ξ21+ξ2,that is, there occurs a Schwarz function κ∈∪,∣κξ∣≤∣ξ∣<1,κ0=0 with the property
(55)Ψξ≔zMξ′Mξ≺1−κξ21+κξ2,which implies that there is ζ,∣ζ∣=r<1 such that
(56)κ2ζ=1−Ψζ1+Ψζ,ζ∈∪.
An operation, one can indicate that(57)∣1−Ψζ1+Ψζ∣=κζ2≤ζ2<1.
Thus, one can inform the recognizing inequality(58)∣Ψz−ζ4+11−ζ4∣≤2ζ21−ζ4.
This implies the consequence result(59)RΨξ≥1−ϱ21+ϱ2.
Next consequence outcomes of the above result can be located in [8, 25, 26] accordingly.Corollary 9.
Ifλ=1 in Theorem 8, then
(60)1+1♭Sq1,k+1MξSq1,kMξ−1≺1+Aξ1+Bξ.Corollary 10.
Ifλ=1,k=1, and q⟶1 in Theorem 8, then
(61)1+1♭Sq1,2MξSq1,1Mξ−1≺1+Aξ1+Bξ.Corollary 11.
Ifq⟶1 in Theorem 8, then
(62)1+1♭Sqλ,k+1MξSqλ,kMξ−1≺1+Aξ1+Bξ.


## 4. Applications

We introduce an application of our outcomes based on finding the outcome of Briot-Bouquet equation (BBE) (see [11] for more information). This category of ODE is an association of ODE whose outcomes are formulas in the complex plane. Existence and uniqueness theorems include the utility of majors and minors (or subordination and superordination concepts) (see [28–31]). Investigation of rational first ODEs in the complex region implies the finding of new transcendental special functions which are now known as symmetric BBE
(63)βγξ+1−βξγξ′γξ=ℏξ,ℏ0=γ0,β∈0,1.By employing theq-differential operator (11), we have the q-formula of BBE
(64)βγξ+1−βξSqλ,kγξ′Sqλ,kγξ=ℏξ,ℏ0=γ0,ξ∈∪.A simple result of (64) can be recognized at β=1. Thus, we investigate the status, γ∈∧ and β=0. The initial condition will be γ0=ℏ0=0.Theorem 12.
Suppose that Equation (64) with β=0 and γ∈∧ with nonnegative coefficients. If ℏξ,ξ∈∪ is convex in ∪, then there occurs a major solution achieving the inequality
(65)Sqλ,kγξ≺ξexp∫0ξhðu−1udu,such that ðξ indicates analytic function in ∪, satisfying the relations ∣ðξ∣<1 and ð0=0.Proof.
In virtue of every conditions indicated by (64), and γξ∈∧, we realize the information
(66)RξSqλ,kγξ′Sqλ,kγξ>0⇔Rξ+∑n=2∞nn,qkλ-1-λ-1nkγnξnξ+∑n=2∞n,qkλ-1-λ-1nkγnξn>0⇔R1+∑n=2∞nn,qkλ-1-λ-1nkγnξn-11+∑n=2∞n,qkλ-1-λ-1nkγnξn-1>0⇔1+∑n=2∞nn,qkλ-1-λ-1nkγn1+∑n=2∞n,qkλ-1-λ-1nkγn>0,where Rξ≠0. Moreover, by the definition of Sqλ,kγξ, we indicate that Sqλ,kγ0=0. Consequently, yields that
(67)ξSqλ,kγξ′Sqλ,kγξ∈P⇒γξ∈Sq∗λ,k,ℏ.
Thus, via Theorem6, we reach the desired outcome in (65).



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