In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.
Citation: Muath Awadalla, Manigandan Murugesan, Manikandan Kannan, Jihan Alahmadi, Feryal AlAdsani. Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems[J]. AIMS Mathematics, 2024, 9(6): 14130-14157. doi: 10.3934/math.2024687
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In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.
Let A denote the class of functions of the form
f(z)=z+∞∑k=2akzk, | (1.1) |
which are analytic in the open disc U={z∈C:|z|<1}. Let S denote the subclass of A consisting of functions that are univalent in U. Also, let Ω be the class of all analytic functions w in U that satisfy the conditions w(0)=0 and |w(z)|<1(z∈U). If f and g are analytic in U, we say that f is subordinate to g, written as f≺g in U or f(z)≺g(z) (z∈U), if there exists w∈Ω such that f(z)=g(w(z)) (z∈U). Furthermore, if the function g(z) is univalent in U, then we have the following equivalence holds (see [4] and [11]):
f(z)≺g(z)⟺f(0)=g(0) and f(U)⊂g(U). |
A function f∈A is said to be in the class of γ− spiral-like functions of order λ in U, denoted by S∗(γ;λ) if
ℜ{eiγzf′(z)f(z)}>λ cosγ (0≤λ<1,|γ|<π2;z∈U). | (1.2) |
The class S∗(γ;λ) was studied by Libera [10] and Keogh and Merkes [9]. Note that
1). S∗(γ;0)=S∗(γ) is the class of spiral-like functions introduced by Špaček [17];
2). S∗(0;λ)=S∗(λ) is the class of starlike functions of order λ;
3). S∗(0;0)=S∗ is the familiar class of starlike functions.
For functions f∈A given by (1.1) and g∈A given by
g(z)=z+∞∑k=2bkzk, | (1.3) |
we define the Hadamard product (or Convolution) of f and g by
(f∗g)(z)=z+∞∑k=2akbkzk. | (1.4) |
Also, for f∈A given by (1.1) and 0<q<1, the Jackson's q-derivative operator or q-difference operator for a function f∈A is defined by (see [1,2,3,6,7,15,16])
Dqf(z):={f′(0)if z=0,f(z)−f(qz)(1−q)zif z≠0. | (1.5) |
From (1.5), we deduce that
Dqf(z)=1+∞∑k=2[k]q akzk−1 (z≠0), | (1.6) |
where the q-integer number [i]q is defined by
[i]q=1−qi1−q=1+q+q2+...+qi−1, | (1.7) |
and
limq→1−Dqf(z)=limq→1−f(z)−f(qz)(1−q)z=f′(z), | (1.8) |
for a function f which is differentiable in a given subset of C.
Next, in terms of the q-generalized Pochhammer symbol ([v]q)n given by
([v]q)n=[v]q [v+1]q [v+2]q ... [v+n−1]q, | (1.9) |
we define the function ϕq(a,c;z) by
ϕq(a,c;z)=z+∞∑k=2([a]q)k−1([c]q)k−1zk (a∈R;c∈R∖Z−0;Z−0={0,−1,−2,...};z∈U). | (1.10) |
Corresponding to the function ϕq(a,c;z), we consider a linear operator Lq(a,c):A→A which is defined by means of the following Hadamard product (or convolution):
Lq(a,c)f(z)=ϕq(a,c;z)∗f(z)=z+∞∑k=2([a]q)k−1([c]q)k−1 ak zk. | (1.11) |
It is easily verified from (1.11) that
qa−1zDq(Lq(a,c)f(z))=[a]q Lq(a+1,c)f(z)−[a−1]qLq(a,c)f(z). | (1.12) |
Moreover, for f∈A, we observe that
1). limq→1−Lq(a,c)f(z)=L(a,c)f(z), where L(a,c) denotes the Carlson-Shaffer operator [5];
2). Lq(δ+1,1)f(z)=Rδqf(z)(δ>0), where Rδq denotes the Ruscheweyh q-derivative of a function f∈A of order δ (see [8]);
3). limq→1−Lq(δ+1,1)f(z)=Rδf(z)(δ>0), where Rδq denotes the Ruscheweyh derivative of order δ (see [14]);
4). Lq(a,a)f(z)=f(z) and Lq(2,1)f(z)=zDqf(z).
Making use of the q-analogue of Carlson-Shaffer operator Lq(a,c), we introduce a new subclass of spiral-like functions.
Definition 1. For 0≤t≤1, 0≤λ<1 and |γ|<π2, we let Saq(γ,λ,t) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:
ℜ{eiγzDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)}>λ cosγ | (1.13) |
(0≤t≤1;0≤λ<1;|γ|<π2;z∈U). |
We note that
1). For t=1, 0≤λ<1 and |γ|<π2, we let Saq(γ,λ,1)=Saq(γ,λ) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:
ℜ{eiγzDq(Lq(a,c)f(z))Lq(a,c)f(z)}>λ cosγ | (1.14) |
(0≤λ<1;|γ|<π2;z∈U). |
2). For t=0, 0≤λ<1 and |γ|<π2, we let Saq(γ,λ,0)=Kaq(γ,λ) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:
ℜ{eiγ Dq(Lq(a,c)f(z))}>λ cosγ | (1.15) |
(0≤λ<1;|γ|<π2;z∈U). |
The object of the present paper is to investigate the coefficient estimates and subordination properties for the class of functions Saq(γ,λ,t). Some interesting consequences of the results are also pointed out.
In this section, we obtain several sufficient conditions for a function f∈A to be in the class Saq(γ,λ,t).
Theorem 1. Let f∈A and let σ be a real number with 0≤σ<1. If
|zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)−1|≤1−σ (z∈U), | (2.1) |
then f∈Saq(γ,λ,t) provided that
|γ|≤cos−1(1−σ1−λ) | (2.2) |
Proof. From (2.1) it follows that
zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)=1+(1−σ)w(z), |
where w(z)∈Ω. We have
ℜ{eiγzDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)}=ℜ{eiγ[1+(1−σ)w(z)]}=cosγ+(1−σ)ℜ{eiγw(z)}≥cosγ−(1−σ)|eiγw(z)|>cosγ−(1−σ)≥λ cosγ |
provided that |γ|≤cos−1(1−σ1−λ). Thus, the proof is completed.
Putting σ=1−(1−λ)cosγ in Theorem 1, we obtain the following result.
Corollary 1. Let f∈A. If
|zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)−1|≤(1−λ)cosγ (z∈U), | (2.3) |
then f∈Saq(γ,λ,t).
In the following theorem, we obtain a sufficient condition for f to be in Saq(γ,λ,t).
Theorem 2. A function f(z) of the form (1.1) is in Saq(γ,λ,t) if
∞∑k=2{([k]q−t)secγ+(1−λ)t}([a]q)k−1([c]q)k−1|ak|≤1−λ. | (2.4) |
Proof. In virtue of Corollary 1, it suffices to show that the condition (2.3) is satisfied. We have
|zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)−1|=|∞∑k=2{[k]q−t}([a]q)k−1([c]q)k−1 ak zk−11+t∞∑k=2([a]q)k−1([c]q)k−1ak zk−1|<∞∑k=2{[k]q−t}([a]q)k−1([c]q)k−1 |ak|1−∞∑k=2t([a]q)k−1([c]q)k−1|ak|. |
The last expression is bounded above by (1−λ)cosγ, if
∞∑k=2{[k]q−t}([a]q)k−1([c]q)k−1 |ak|≤(1−λ)cosγ{1−∞∑k=2t([a]q)k−1([c]q)k−1|ak|} |
which is equivalent to
∞∑k=2{([k]q−t)secγ+(1−λ)t}([a]q)k−1([c]q)k−1|ak|≤1−λ. |
This completes the proof of the Theorem 2.
Putting t=1 in Theorem 2, we obtain the following corollary.
Corollary 2. A function f(z) of the form (2.1) is in Saq(γ,λ) if
∞∑k=2{([k]q−1)secγ+1−λ}([a]q)k−1([c]q)k−1|ak|≤1−λ. | (2.5) |
Putting t=0 in Theorem 2, we obtain the following corollary.
Corollary 3. A function f(z) of the form (2.1) is in Kaq(γ,λ) if
∞∑k=2[k]qsecγ([a]q)k−1([c]q)k−1|ak|≤1−λ. | (2.6) |
Before stating and proving our subordination result for the class Saq(γ,λ,t), we need the following definitions and a lemma due to Wilf [19].
Definition 2 [19]. A sequence {bk}∞k=1 of complex numbers is said to be a subordinating factor sequence if, whenever f(z)=z+∞∑k=2akzk is regular, univalent and convex in U, we have
∞∑k=1ak bk ≺f(z) (a1=1;z∈U). | (3.1) |
Lemma 1 [19]. The sequence {bk}∞k=1 is a subordinating factor sequence if and only if
ℜ{1+2∞∑k=1bk zk}>0 (z∈U). | (3.2) |
Theorem 3. Let f∈Saq(γ,λ,t) satisfy the coefficient inequality (2.4) with a≥c>0 and let g(z) be any function in the usual class of convex functions C, then
{([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q](f∗g)(z)≺g(z) | (3.3) |
and
ℜ{f(z)}>−1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q{([2]q−t)secγ+(1−λ)t}[a]q[c]q. | (3.4) |
The constant factor {([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q] in (3.3) cannot be replaced by a larger number.
Proof. Let f∈Saq(γ,λ,t) satisfy the coefficient inequality (2.4) and suppose that
g(z)=z+∞∑k=2bkzk∈C. |
Then, by Definition 2, the subordination (3.3) of our theorem will hold true if the sequence
{{([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q]ak}∞k=1 |
is a subordinating factor sequence, with b1=1. In view of Lemma 1, it is equivalent to the inequality
ℜ{1+∞∑k=1{([2]q−t)secγ+(1−λ)t}[a]q[c]q1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]qak zk}>0 (z∈U). | (3.5) |
By noting the fact that {([k]q−t)secγ+(1−λ)t}([a]q)k−1(1−λ)([c]q)k−1 is an increasing function for k≥2 and a≥c>0. In view of (2.4), when |z|=r<1, we obtain
ℜ{1+{([2]q−t)secγ+(1−λ)t}[a]q[c]q1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q∞∑k=1ak zk} |
=ℜ{1+{([2]q−t)secγ+(1−λ)t}[a]q[c]q1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]qz+∞∑k=2{([2]q−t)secγ+(1−λ)t}[a]q[c]qak zk1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q} |
≥1−{([2]q−t)secγ+(1−λ)t}[a]q[c]q1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]qr−∞∑k=2{([k]q−t)secγ+(1−λ)t}([a]q)k−1([c]q)k−1|ak| rk1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q |
≥1−{([2]q−t)secγ+(1−λ)t}[a]q[c]q1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]qr−1−λ1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]qr=1−r>0 (|z|=r<1). |
This evidently proves the inequality (3.5) and hence also the subordination result (3.3) asserted by Theorem 3. The inequality (3.4) follows from (3.3) by taking
g(z)=z1−z=z+∞∑k=2zk∈C. |
The sharpness of the multiplying factor in (3.3) can be established by considering a function
F(z)=z−1−λ{([2]q−t)secγ+(1−λ)t}[a]q[c]qz2. |
Clearly F∈Saq(γ,λ,t) satisfy (2.4). Using (3.3) we infer that
{([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q]F(z)≺z1−z, |
and it follows that
min|z|≤r{{([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q]ℜ{F(z)}}=−12. |
This shows that the constant {([2]q−t)secγ+(1−λ)t}[a]q[c]q2[1−λ+{([2]q−t)secγ+(1−λ)t}[a]q[c]q] cannot be replaced by any larger one.
For t=1 in Theorem 3, we state the following corollary.
Corollary 4. Let f∈Saq(γ,λ) satisfy the coefficient inequality (2.5) with a≥c>0 and g∈C, then
{qsecγ+1−λ}[a]q[c]q2[1−λ+{qsecγ+1−λ}[a]q[c]q](f∗g)(z)≺g(z) | (3.6) |
and
ℜ{f(z)}>−1−λ+{qsecγ+1−λ}[a]q[c]q{qsecγ+1−λ}[a]q[c]q. | (3.7) |
The constant factor {qsecγ+1−λ}[a]q[c]q2[1−λ+{qsecγ+(1−λ)}[a]q[c]q] in (3.6) cannot be replaced by a larger number.
Taking t=0 in Theorem 3, we state the next corollary.
Corollary 5. Let f∈Kaq(γ,λ) satisfy the coefficient inequality (2.6) with a≥c>0 and g∈C, then
(1+q)secγ[a]q[c]q2[1−λ+(1+q)secγ[a]q[c]q](f∗g)(z)≺g(z) | (3.8) |
and
ℜ{f(z)}>−1−λ+(1+q)secγ[a]q[c]q(1+q)secγ[a]q[c]q. | (3.9) |
The constant factor (1+q)secγ[a]q[c]q2[1−λ+(1+q)secγ[a]q[c]q] in (3.8) cannot be replaced by a larger number.
The Fekete-Szegö problem consists in finding sharp upper bounds for the functional |a3−μa22| for various subclasses of A (see [13] and [18]). In order to obtain sharp upper-bounds for |a3−μa22| for the class Saq(γ,λ,t) the following lemma is required (see, e.g., [12, p.108]).
Lemma 2. Let the function w∈Ω be given by
w(z)=∞∑k=1wk zk (z∈U). |
Then
|w1|≤1, |w2|≤1−|w1|2, | (4.1) |
and
|w2−s w21|≤max{1,|s|}, | (4.2) |
for any complex number s. The functions w(z)=z and w(z)=z2or one of their rotations show that both inequalities (4.1) and (4.2) are sharp.
For the constants λ, γ with 0≤λ<1 and |γ|<π2 denote
Pλ,γ(z)=1+e−iγ(e−iγ−2λcosγ)z1−z (z∈U). | (4.3) |
The function Pλ,γ(z) maps the open unit disk U onto the half-plane Hλ,γ={w∈C:ℜ{eiγw}>λcosγ}. If
Pλ,γ(z)=1+∞∑k=1pkzk (z∈U), |
then it is easy to check that
pk=2e−iγ(1−λ)cosγ (k≥1). | (4.4) |
First we obtain sharp upper-bounds for the Fekete-Szegö functional |a3−μa22| with μ real parameter.
Theorem 4. Let f∈Saq(γ,λ,t) be given by (1.1) and let μ be a real number. Then
|a3−μa22|≤{2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[1+2(1−λ)t1+q−t −μ2(1−λ)(1+q+q2−t)([a]q)2([c]q)2(1+q−t )2([a]q)2([c]q)2](μ≤σ1)2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2(σ1≤μ≤σ2)2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[−1−2(1−λ)t1+q−t +μ2(1−λ)(1+q+q2−t)([a]q)2([c]q)2(1+q−t )2([a]q)2([c]q)2](μ≥σ2) | (4.5) |
where
σ1=t(1+q−t) ([a]q)2([c]q)2(1+q+q2−t)([c]q)2([a]q)2 | (4.6) |
σ2= (1+q−t)(1+q−tλ)([a]q)2([c]q)2(1−λ)(1+q+q2−t)([a]q)2 ([c]q)2 | (4.7) |
and all estimates are sharp.
Proof. Suppose that f∈Saq(γ,λ,t) is given by (1.1). Then, from the definition of the class Saq(γ,λ,t), there exists w∈Ω,
w(z)=w1z+w2z2+w3z3+... |
such that
zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)=Pλ,γ(w(z)) (z∈U). | (4.8) |
We have
zDq(Lq(a,c)f(z))(1−t) z+t Lq(a,c)f(z)=1+(1+q−t)[a]q[c]qa2 z |
+{(1+q+q2−t)([a]q)2([c]q)2 a3−t(1+q−t)([a]q)2([c]q)2 a22}z2+... | (4.9) |
Set
Pλ,γ(z)=1+p1z+p2z2+p3z3+... . |
From (4.4) we have
p1=p2=2e−iγ(1−λ)cosγ. |
Equating the coefficients of z and z2 on both sides of (4.8) and using (4.9), we obtain
a2=p1 [c]q(1+q−t) [a]qw1 |
and
a3=([c]q)2(1+q+q2−t)([a]q)2[p1w2+(p2+t (1+q−t) p21)w21] |
and thus we obtain
a2=2e−iγ(1−λ)cosγ [c]q(1+q−t) [a]qw1 | (4.10) |
and
a3=2e−iγ(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[w2+(1+2te−iγ(1−λ)cosγ 1+q−t )w21]. | (4.11) |
It follows
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[|w2|+|1+2e−iγ(1−λ)cosγ1+q−t (t −μ(1+q+q2−t)([a]q)2([c]q)2(1+q−t) ([a]q)2([c]q)2)||w1|2] |
Making use of Lemma 2 we have
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[1+(|1+2e−iγ(1−λ)cosγ1+q−t (t −μ(1+q+q2−t)([a]q)2 ([c]q)2([c]q)2(1+q−t) ([a]q)2)|−1)|w1|2] |
or
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[1+(√1+M(M+2)cos2γ−1)|w1|2], | (4.12) |
where
M=2(1−λ)1+q−t (t −μ(1+q+q2−t)([a]q)2 ([c]q)2([c]q)2(1+q−t) ([a]q)2). | (4.13) |
Denote by
F(x,y)=[1+(√1+M(M+2)x2−1)y2] |
where x=cosγ, y=|w1| and (x,y):[0,1]×[0,1].
Simple calculation shows that the function F(x,y) does not have a local maximum at any interior point of the open rectangle (0,1)×(0,1). Thus, the maximum must be attained at a boundary point. Since F(x,0)=1, F(0,y)=1 and F(1,1)=|1+M|, it follows that the maximal value of F(x,y) may be F(0,0)=1 or F(1,1)=|1+M|. Therefore, from (4.12) we obtain
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2max{1,|1+M|}, | (4.14) |
where M is given by (4.13). Consider first the case |1+M|≥1. If μ≤σ1, where σ1 is given by (4.6), then M≥0 and from (4.14) we obtain
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[1+2(1−λ)t1+q−t −μ2(1−λ)(1+q+q2−t)([a]q)2 ([c]q)2(1+q−t )2([a]q)2([c]q)2] |
which is the first part of the inequality (4.5). If μ≥σ2, where σ2 is given by (4.7), then M≤−2 and it follows from (4.14) that
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2[−1−2(1−λ)t1+q−t +μ2(1−λ)(1+q+q2−t)([a]q)2 ([c]q)2(1+q−t )2([a]q)2([c]q)2] |
and this is the third part of (4.5).
Next, suppose σ1≤μ≤σ2. Then, |1+M|≤1 and thus, from (4.14) we obtain
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2 |
which is the second part of the inequality (4.5). In view of Lemma 2, the results are sharp for w(z)=z and w(z)=z2 or one of their rotations.
For t=1 in Theorem 4, we state the following corollary.
Corollary 6. Let f∈Saq(γ,λ) be given by (1.1) and let μ be a real number. Then
|a3−μa22|≤{2(1−λ)cosγ([c]q)2q(1+q)([a]q)2[1+2(1−λ)q−μ2(1−λ)(1+q)([a]q)2([c]q)2q([a]q)2([c]q)2](μ≤σ3)2(1−λ)cosγ([c]q)2q(1+q)([a]q)2(σ3≤μ≤σ4)2(1−λ)cosγ([c]q)2q(1+q)([a]q)2[−1−2(1−λ)q+μ2(1−λ)(1+q)([a]q)2([c]q)2q([a]q)2([c]q)2](μ≥σ4) |
where
σ3=([a]q)2([c]q)2(1+q)([c]q)2([a]q)2, σ4= (1+q−λ)([a]q)2([c]q)2(1−λ)(1+q)([a]q)2 ([c]q)2 |
and all estimates are sharp.
Taking t=0 in Theorem 4, we state the next corollary.
Corollary 7. Let f∈Kaq(γ,λ) be given by (1.1) and let μ be a real number. Then
|a3−μa22|≤{2(1−λ)cosγ([c]q)2(1+q+q2)([a]q)2[1−μ2(1−λ)(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2](μ≤0)2(1−λ)cosγ([c]q)2(1+q+q2)([a]q)2(0≤μ≤σ5)2(1−λ)cosγ([c]q)2(1+q+q2)([a]q)2[−1+μ2(1−λ)(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2](μ≥σ5) |
where
σ5= (1+q)2([a]q)2([c]q)2(1−λ)(1+q+q2)([a]q)2 ([c]q)2 |
and all estimates are sharp.
We consider the Fekete-Szegö problem for the class Saq(γ,λ,t) with μ complex parameter.
Theorem 5. Let f∈Saq(γ,λ,t) be given by (1.1) and let μ be a complex number. Then,
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2max{1,|2(1−λ)cosγ 1+q−t (μ(1+q+q2−t)([a]q)2([c]q)2(1+q−t)([a]q)2([c]q)2−t)−eiγ|}. | (4.15) |
The result is sharp.
Proof. Assume that f∈Saq(γ,λ,t). Making use of (4.10) and (4.11) we obtain
|a3−μa22|=2(1−λ)cosγ([c]q)2(1+q+q2−t)([a]q)2|w2−[2e−iγ(1−λ)cosγ 1+q−t (μ(1+q+q2−t)([a]q)2([c]q)2(1+q−t)([a]q)2([c]q)2−t)−1]w21| |
The inequality (4.15) follows as an application of Lemma 2 with
s=2e−iγ(1−λ)cosγ 1+q−t (μ(1+q+q2−t)([a]q)2([c]q)2(1+q−t)([a]q)2([c]q)2−t)−1. |
For t=1 in Theorem 5, we state the following corollary.
Corollary 8. Let f∈Saq(γ,λ) be given by (1.1) and let μ be a complex number. Then,
|a3−μa22|≤2(1−λ)cosγ([c]q)2q(1+q)([a]q)2max{1,|2(1−λ)cosγ q (μ(1+q)([a]q)2([c]q)2([a]q)2([c]q)2−1)−eiγ|}. |
The result is sharp.
Taking t=0 in Theorem 5, we state the next corollary.
Corollary 9. Let f∈Kaq(γ,λ) be given by (1.1) and let μ be a complex number. Then,
|a3−μa22|≤2(1−λ)cosγ([c]q)2(1+q+q2)([a]q)2max{1,|μ2(1−λ)cosγ(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2−eiγ|}. |
The result is sharp.
Utilizing the concepts of quantum calculus, we defined new subclass of analytic functions associated with q-analogue of Carlson-Shaffer operator. For this subclass we investigated some useful results such as coefficient estimates, subordination properties and Fekete-Szegö problem. Their are some problems open for researchers such as distortion theorems, closure theorems, convolution propertiies and radii problems. Moreover, these results can be extended to multivalent functions and meromophic functions.
The authors are thankful to the referees for their valuable comments which helped in improving the paper.
The authors declare that they have no competing interests.
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