In this work, a new class of bi-univalent functions \(I^{n+1}_{\Gamma_m,\lambda}(x,z)\) is defined by means of subordination. Upper bounds for some initial coefficients and the Fekete-Szegö functional of functions in the new class were obtained.
Let \(A\) denote the class of functions that are analytic within the unit disk, defined as \[U = \{ z \in \mathbb{C} : |z| < 1 \}.\] Furthermore, let \(S\) be a subclass of \(A\), comprising functions that are univalent in \(U\) and adhere to the normalization \[\label{eq1} f(z) = z + \sum_{k=2}^{\infty} a_k z^k. \tag{1}\]
The Koebe one-quarter theorem [1] assures that for every univalent function \(f \in S\), its image in \(U\) encompasses a disk with a minimum radius of \(\frac{1}{4}\). Consequently, every univalent function \(f\) possesses an inverse \(f^{-1}\) satisfying \[f^{-1}(f(z)) = z \quad (z \in U), \quad f(f^{-1}(w)) = w \quad (|w| < r_0(f)), \quad r_0(f) \geq \frac{1}{4}. \tag{2}\]
Definition 1. A function \(f \in S\) is termed bi-univalent in \(U\) if \(f\) is univalent in \(U\) and its inverse \(f^{-1}\) extends univalently to \(U\).
The class of bi-univalent functions in \(U\) is denoted by \(\Sigma\). Since each function \(f \in \Sigma\) can be expressed by the Maclaurin series (1), it follows that its inverse \(g = f^{-1}\) can be expanded as \[g(w) = w – a_2 w^2 + (2a_2^2 – a_3) w^3 + \ldots. \tag{3}\]
Recent studies have focused on several classes of bi-univalent functions (see, for instance, references [2– 4]).
Definition 2. For functions \(f\) and \(g\) in the class \(A\), \(f\) is said to be subordinate to \(g\) if there exists a Schwarz function \(w(z)\) with \(w(0) = 0\) and \(|w(z)| < 1\), such that \(f(z) = g(w(z))\).
In particular, if \(g\) is univalent in \(U\), subordination implies \[f(0) = g(0) \quad \text{and} \quad f(U) \subseteq g(U).\]
We denote by \[S^* = \{ f \in S : \Re \left( \frac{zf'(z)}{f(z)} \right) > 0, \, z \in U \}\] and \[K = \{ f \in S : \Re \left( 1 + \frac{zf''(z)}{f'(z)} \right) > 0, \, z \in U \}.\]
The classes \(S^*\) and \(K\) are referred to as the classes of starlike and convex functions, respectively.
A noteworthy concept in the theory of univalent functions, recently revisited in the context of singularity theory[5] and power series with integral coefficients, is the Hankel determinant \(H_{\delta}(n)\) for functions \(f(z) \in A\) with the form (1). For \(\delta \geq 1\) and \(m \geq 1\), \[H_{\delta}(n) = \begin{vmatrix} a_{m} & a_{m+1} & \cdots & a_{m+\delta-1} \\ a_{m+1} & a_{m+2} & \cdots & a_{m+\delta} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m+\delta-1} & a_{m+\delta} & \cdots & a_{m+2(\delta-1)} \end{vmatrix}. \tag{4}\]
This determinant has been extensively investigated in the literature [6– 8]. The Hankel determinant \(H_2(1) = a_3 – a_2^2\) is well known as the Fekete-Szegö functional. Identifying the upper bound for \(\lambda_{\sigma}(f) = |a_3 – \sigma a_2^2|\) over the class \(S\), a generalization of \(H_2(1)\), is recognized as the Fekete-Szegö problem, where \(\sigma\) is a real or complex number.
Using the Loewner method, Fekete and Szegö demonstrated in [9] that \[\max_{f \in S} \lambda_{\sigma}(f) = \begin{cases} 1 + 2\exp\left(\frac{-2\sigma}{1-\sigma}\right), & \sigma \in [0,1], \\ 1, & \sigma = 1. \end{cases}\]
In 1969, the Fekete-Szegö problem for the classes of starlike functions \(S^*\) and convex functions \(K\) was investigated by Koegh and Merkes [10].
Recently, Hörcum and Gokcen [11] considered the Horadam polynomials \(J_m(x)\), defined by the recurrence relation \[J_m(x) = pxJ_{m-1}(x) + qJ_{m-2}(x), \tag{5}\] where \(m \in \mathbb{N} \setminus \{1, 2\}\), \(\mathbb{N} = \{1, 2, 3, \ldots\}\), and \(a\), \(h\), \(p\), \(q\) are specific real constants.
For particular cases:
For \(a = h = p = q = 1\), we obtain the Fibonacci polynomial \(F_n(x)\).
For \(a = 1\), \(h = p = 2\) and \(q = -1\), we acquire the Chebyshev polynomial \(U_n(x)\) of the second kind.
Definition 3. [12] Let \(f \in A\). Define the operator \(D^n(\mu,\beta,t)f(z)\) as follows: \[\begin{aligned} D^n(\mu,\beta,t)f(z): A &\rightarrow A, \quad n \in N_0 = N \cup \{0\}, \\ D^0(\mu,\beta,t)f(z) &= f(z), \\ D^1(\mu,\beta,t)f(z) &= tzf'(z) – z(\beta – \mu)t + (1 + (\beta – \mu – 1)t)f(z), \\ D^n(\mu,\beta,t)f(z) &= zD_t[D^{n-1}(\mu,\beta,t)f(z)], \end{aligned}\] where \(D_tf(z) = 1 + \sum_{k=2}^{\infty}[1 + (k + \beta – \mu – 1)t]a_kz^k\). If \(f(z)\) is expressed as in equation (1), then \[D^n(\mu,\beta,t)f(z) = z + \sum_{k=2}^{\infty}[1 + (k + \beta – \mu – 1)t]^n a_k z^k,\] with \(0 \leq \mu \leq \beta\) and \(t \geq 0\).
Remark 1.
Definition 4. A function \(f(z) \in A\) belongs to the class \(I^{n+1}_{\Gamma_m,\lambda}(x,z)\) if \[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)} + \left(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)}\right)^{\frac{1}{\lambda}}\right] \prec \prod(x,z) + 1 – a, \tag{6}\] and similarly for \(g(w)\), \[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)} + \left(\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)}\right)^{\frac{1}{\lambda}}\right] \prec \prod(x,w) + 1 – a, \tag{7}\] where \(n \in N_0 = N \cup \{0\}\), \(\Gamma_m = 1 + (m + \beta – \mu)t\), \(m \in N\), \(\mu \in [0, \beta]\), \(t \geq 0\), \(0 < \lambda \leq 1\), and \(f(z)\), \(g(w)\) are as previously defined. Here, \(D^n_{\Gamma_m}f(z)\) denotes the Opoola differential operator, \(\prod(x,z) = \sum_{m=1}^{\infty} j_m(x)z^{m-1}\), and \(z, w \in U\).
Remark 2.
For \(n = 0\), we obtain the class \(W_\Sigma(\mu;x)\) of functions satisfying: \[\frac{1}{2}\left[\frac{zf'(z)}{f(z)} + \left(\frac{zf'(z)}{f(z)}\right)^{\frac{1}{\lambda}}\right] \prec \prod(x,z) + 1 – a, \tag{8}\] and similarly for \(g(w)\). When \(n = 0\) and \(\lambda = 1\), this class reduces to \(W_\Sigma(x)\), comprising functions \(f \in A\) satisfying certain criteria, as studied by Srivastava et al. [15].
For \(n = 0\), \(a = 1\), and other specific parameters, the class \(S_\Sigma(\mu;t)\), as studied by Altinkaya and Yakin [16], is obtained.
Let \(P\) denote the class of functions with positive real part. A function \(p \in P\) implies that \[p(z) = 1 + b_1z + b_2z^2 + b_3z^3 + \ldots, \quad z \in U, \tag{9}\] with \(\Re p(z) > 0\). It is known that \[p(z) = \frac{w(z) – 1}{w(z) + 1}, \tag{10}\] where \(w(z)\) is a Schwarz function.
Lemma 1. [17] If \(p(z) = 1 + b_1z + b_2z^2 + b_3z^3 + \ldots\) is in \(P\), then \(|b_m| \leq 2\).
Lemma 1 [17]:
Lemma 2. [17] If \(p(z) = 1 + b_1z + b_2z^2 + b_3z^3 + \ldots\) is in \(P\), then \[\left|b_2 – \sigma b_1^2\right| \leq \begin{cases} 2 – \sigma|b_1|^2, & \sigma \leq \frac{1}{2}, \\ 2 – (1 – \sigma)|b_1|^2, & \sigma \geq \frac{1}{2}. \end{cases} \tag{11}\]
Theorem 1. . Let \(f(z) \in I^{n+1}_{\Gamma_m,\lambda}(x,z)\), then \[\label{important} |a_2| \leq \frac{2\lambda\sqrt{|hx|}|hx|}{\sqrt|{[2\lambda(1+\lambda)\Gamma_2^{n}(\Gamma_2 -1)+\Gamma_1^{2n}(4\lambda\Gamma_1-1-3\lambda)](hx)^2+(hx-(phx^2+qa))(1+\lambda)^2(\Gamma_1-1)^2\Gamma_1^{2n}|}} \tag{12}\] \[|a_3|\leq \frac{2\lambda|hx|(1+2\lambda)}{\Gamma_2^n(\Gamma_2-1)(1+\lambda)}+\frac{4\lambda^2(hx)^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2} \tag{13}\]
Proof. Let \(f(z)\in
I^{n+1}_{\Gamma_m,\lambda}(x,z)\).By definition of Subordination,
there exists functions \(\Theta(z)\)
and \(\Phi(z)\), \(\Theta(0)=\Phi(0)=0\) and \(|\Theta(z)|<1\) and \(|\Phi(w)|<1\) \(\forall z, w \in U\) such that
\[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)}
+
(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)})^{\frac{1}{\lambda}}\right]
= \prod(x,\Theta(z))+1-a \tag{14}\] and
\[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)}
+
(\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)})^{\frac{1}{\lambda}}\right]
= \prod(x,\Phi(w))+1-a \tag{15}\] From \(15\), \[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)}
+
(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)})^{\frac{1}{\lambda}}\right]=
\sum_{m=1}^{\infty} j_m(x)(\Theta(z))^{m-1}+1-a\] \[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)}
+
(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)})^{\frac{1}{\lambda}}\right]=j_1(x)+j_2(x)\Theta(z)+j_3(x)[\Theta(z)]^2+j_4(x)[\Theta(z)]^3+…+1-a \tag{16}\]
Also, from \(16\)
\[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)}
+
(\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)})^{\frac{1}{\lambda}}\right]=j_1(x)+j_2(x)\Phi(w)+j_3(x)[\Phi(w)]^2+j_4(x)[\Phi(w)]^3+…+1-a \tag{17}\]
From equations \(11\) and \(12\)
\[\Theta(z)= \frac{p(z)-1}{p(z)+1}\]
\[=(b_1 z+b_2 z^2+b_3z^3+b_4z^4+…)(2+b_1
z+b_2 z^2+b_3z^3+b_4z^4+…)^{-1}\] \[\Theta(z)= \frac{1}{2}[b_1
z+(b_2-\frac{b_1^{2}}{2})
z^2+(b_3-b_1b_2+\frac{b_1^{3}}{4})z^3+…] \tag{18}\] \[^2= \frac{1}{4}[b_1^2
z^2+2(b_1b_2-\frac{b_1^{3}}{2})
z^3+(b_1b_3-b_1^2b_2+\frac{2b_1^{4}}{4}-b_1^3b_2+b_1^2b_2^2)z^4+…] \tag{19}\]
\[^3=
\frac{1}{8}[b_1^3z^3+3(b_1^2b_2-\frac{b_1^{4}}{2}) z^4+…] \tag{20}\]
\[\Phi(w)=
\frac{\rho(w)-1}{\rho(w)+1}\] \[=(d_1
w+d_2 w^2+d_3w^3+b_4z^4+…)(2+d_1 w+d_2
w^2+d_3w^3+b_4w^4+…)^{-1}\] \[\Phi(w)= \frac{1}{2}[d_1
w+(d_2-\frac{d_1^{2}}{2})
w^2+(d_3-d_1d_2+\frac{d_1^{3}}{4})w^3+…] \tag{21}\] \[^2= \frac{1}{4}[d_1^2
w^2+2(d_1d_2-\frac{d_1^{3}}{2})
w^3+(d_1d_3-d_1^2d_2+\frac{2d_1^{4}}{4}-d_1^3d_2+d_1^2d_2^2)w^4+…] \tag{22}\]
\[^3=
\frac{1}{8}[d_1^3w^3+3(d_1^2d_2-\frac{d_1^{4}}{2}) w^4+…] \tag{23}\]
Substituting \(19\), \(20\) and \(21\) into \(RHS\) of \(17\), we have
\[\begin{gathered}
\small
\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)} +
(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)})^{\frac{1}{\lambda}}\right]=
1+j_2(x)\frac{b_1}{2}z+
\frac{1}{2}[j_3(x)\frac{b_1^2}{2}+j_2(x)(b_2-\frac{b_1^2}{2})] z^2\\+
\frac{1}{2}[j_4(x)\frac{b_1^3}{4}+j_3(x)(b_1b_2-\frac{b_1^3}{2})+j_2(x)(b_3-b_1b_2+\frac{b_1^3}{4})]z^3+…
\end{gathered} \tag{24}\] Substituting \(22\), \(23\) and \(24\) into \(RHS\) of \(18\), we have
\[\begin{gathered}
\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)} +
(\frac{D^{n+1}_{\Gamma_m}g(w)}{D^{n}_{\Gamma_m}g(w)})^{\frac{1}{\lambda}}\right]=
1+j_2(x)\frac{d_1}{2}z+
\frac{1}{2}[j_3(x)\frac{d_1^2}{2}+j_2(x)(d_2-\frac{d_1^2}{2})] z^2\\+
\frac{1}{2}[j_4(x)\frac{d_1^3}{4}+j_3(x)(d_1d_2-\frac{d_1^3}{2})+j_2(x)(d_3-d_1d_2+\frac{d_1^3}{4})]z^3+…
\end{gathered} \tag{25}\] Considering the \(LHS\) of \(17\)
\[\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)}
+
(\frac{D^{n+1}_{\Gamma_m}f(z)}{D^{n}_{\Gamma_m}f(z)})^{\frac{1}{\lambda}}\right]=
\frac{1}{2}\left[\frac{D^{n+1}_{\Gamma_m}f(z)(D^{n}_{\Gamma_m}f(z))^{\frac{1}{\lambda}}+(D^{n+1}_{\Gamma_m}f(z))^{\frac{1}{\lambda}}D^{n}_{\Gamma_m}f(z)}{(D^{n}_{\Gamma_m}f(z))^{\frac{1+\lambda}{\lambda}}}\right] \tag{26}\]
\[D^{n+1}_{\Gamma_m}f(z)=z+\sum_{k=2}^{\infty}
[1+(k+\beta-\mu-1)t]^{n+1}a_kz^k=z+\Gamma_1^{n+1}a_2z^2+\Gamma_2^{n+1}a_3z^3+\Gamma_3^{n+1}a_4z^4+… \tag{27}\]
\[^{\frac{1}{\lambda}}=[z+\Gamma_1^{n+1}a_2z^2+\Gamma_2^{n+1}a_3z^3+\Gamma_3^{n+1}a_4z^4+…]^{\frac{1}{\lambda}} \tag{28}\]
\[D^{n}_{\Gamma_m}f(z)=z+\sum_{k=2}^{\infty}
[1+(k+\beta-\mu-1)t]^{n}a_kz^k=z+\Gamma_1^{n}a_2z^2+\Gamma_2^{n}a_3z^3+\Gamma_3^{n}a_4z^4+… \tag{29}\]
\[^{\frac{1}{\lambda}}=[z+\Gamma_1^{n}a_2z^2+\Gamma_2^{n}a_3z^3+\Gamma_3^{n}a_4z^4+…]^{\frac{1}{\lambda}} \tag{30}\]
\[^{\frac{1+\lambda}{\lambda}}=[z+\Gamma_1^{n}a_2z^2+\Gamma_2^{n}a_3z^3+\Gamma_3^{n}a_4z^4+…]^{\frac{1+\lambda}{\lambda}} \tag{31}\]
Equation \(28\) multiplied by \(31\), gives
\[\begin{gathered}
^{\frac{1}{\lambda}}=z^{\frac{1}{\lambda}}+
(\frac{1}{\lambda}\Gamma_1^na_2+\Gamma_1^{n+1}a_2)z^{\frac{1+2\lambda}{\lambda}}\\+(\frac{1}{\lambda}\Gamma_2^{n}a_3+\frac{1}{\lambda}\Gamma_1^{2n+1}a_2^2+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2+\Gamma_2^{n+1}a_3)z^{\frac{1+3\lambda}{\lambda}}\\+(\frac{1}{\lambda}\Gamma_3^na_4+\frac{1-\lambda}{\lambda^2}\Gamma_1^n\Gamma_2^na_2a_3+\frac{(1-\lambda)(1-2\lambda)\Gamma_1^{3n+3}a_2^3}{6\lambda^3}\\+\frac{1}{\lambda}\Gamma_1^{n+1}\Gamma_2^{n}a_2a_3+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{3n+1}a_2^3\\+\frac{1}{\lambda}\Gamma_1^{n}\Gamma_2^{n+1}a_2a_3+\Gamma_3{n+1}a_4)z^{\frac{1+4\lambda}{\lambda}}+…
\end{gathered} \tag{32}\] Equation \(29\)
multiplied by \(30\), gives \[\begin{gathered}
^{\frac{1}{\lambda}}[D^{n}_{\Gamma m}
f(z)]=z^{\frac{1+\lambda}{\lambda}}+(\frac{1}{\lambda}\Gamma_1^{n+1}a_2
+ \Gamma_1^n a_2)z^{\frac{1+2\lambda}{\lambda}}+\\
(\frac{1}{\lambda}\Gamma_2^{n+1}a_3+\frac{1}{\lambda}\Gamma_1^{2n+1}+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n+2}a_2^2+\Gamma_2^n
a_3)z^{\frac{1+3\lambda}{\lambda}}\\+(\frac{1}{\lambda}\Gamma_1^{n+1}a_4+\frac{1-\lambda}{\lambda^2}\Gamma_1^{n+1}\Gamma_2^{n+1}a_2a_3\\+\frac{(1-\lambda)(1-2\lambda)}{6\lambda^3}\Gamma_1^{3n+3}a_2^3+\frac{1}{\lambda}\Gamma_2^{n+1}\Gamma_1^n
a_2a_3+
\frac{1-\lambda}{2\lambda^2}\Gamma_1^{3n+2}a_2^3+\frac{1}{\lambda}\Gamma_1^{n+1}\Gamma_2^n
a_2a_3+\Gamma_3^n a_4)z^{\frac{1+4\lambda}{\lambda}}+….
\end{gathered} \tag{33}\]
\[\begin{gathered} \begin{split} 2(D^n_\Gamma m f(z))^\frac{1+\lambda}{\lambda} &= 2z^{\frac{1+\lambda}{\lambda}}+\frac{2(1+\lambda)}{\lambda}\Gamma_1^n a_2z^{\frac{1+2\lambda}{\lambda}}\\ &\quad+2\left(\frac{1+\lambda}{\lambda}\Gamma_2^na_3+\frac{1+\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2\right)z^{\frac{1+3\lambda}{\lambda}}\\ &\quad+2\left(\frac{1+\lambda}{\lambda}\Gamma_3^na_4+\frac{1+\lambda}{\lambda^2}\Gamma_1^n\Gamma_2^na_2a_3+\frac{(1+\lambda)(1-\lambda)}{6\lambda^3}\Gamma_1^{3n}a_2^3\right)z^{\frac{1+4\lambda}{\lambda}}+\dots \end{split} \end{gathered} \tag{34}\]
Substituting \(33\), \(34\) and \(35\) into \(25\) and simplifying further, we have
\[\begin{gathered}
1+[\frac{1+\lambda}{2\lambda}\Gamma_1^n(\Gamma_1+1)]a_2z+\frac{1}{2}[(\frac{1+\lambda}{\lambda}\Gamma_2^n(\Gamma_2+1))a_3+\frac{2}{\lambda}\Gamma_1^{2n+1}a_2^2+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n}(\Gamma_1+1)a_2^2]z^2+…=
1\\+[\frac{1+\lambda}{\lambda}\Gamma_1^na_2+\frac{j_2(x)}{2}b_1]z\\+[\frac{1+\lambda}{\lambda}\Gamma_2^na_3+\frac{1+\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2+\frac{1+\lambda}{2\lambda}j_2(x)b_1+j_3(x)\frac{b_1^2}{4}+\frac{j_2(x)}{2}(b_2-\frac{b_1^2}{2})]z^2+…
\end{gathered} \tag{35}\] Also, \(18\)
becomes,
\[\begin{gathered}
1-[\frac{1+\lambda}{2\lambda}\Gamma_1^n(\Gamma_1+1)]a_2z+\frac{1}{2}[(\frac{1+\lambda}{\lambda}\Gamma_2^n(\Gamma_2+1))(2a_2^2-a_3)+\frac{2}{\lambda}\Gamma_1^{2n+1}a_2^2+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n}(\Gamma_1+1)a_2^2]z^2+…=\\
1+[\frac{1+\lambda}{\lambda}\Gamma_1^na_2+\frac{j_2(x)}{2}d_1]z\\+[\frac{1+\lambda}{\lambda}\Gamma_2^na_3+\frac{1+\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2+\frac{1+\lambda}{2\lambda}j_2(x)d_1+j_3(x)\frac{d_1^2}{4}+\frac{j_2(x)}{2}(d_2-\frac{d_1^2}{2})]z^2+…
\end{gathered} \tag{36}\] Comparing the coefficients of \(z\) in \(36\), we have
\[[\frac{1+\lambda}{2\lambda}\Gamma_1^n(\Gamma_1+1)]a_2
= \frac{1+\lambda}{\lambda}\Gamma_1^na_2+\frac{j_2(x)}{2}b_1\]
\[\frac{1+\lambda}{2\lambda}\Gamma_1^n(\Gamma_1-1)a_2=
j_2(x)\frac{b_1}{2} \tag{37}\] \[a_2=\frac{\lambda
j_2(x)b_1}{(1+\lambda)\Gamma_1^n(\Gamma_1-1)} \tag{38}\] \[
[\frac{1+\lambda}{\lambda}\Gamma^n_2(\Gamma_2+1)]a_3+\frac{2}{\lambda}\Gamma_1^{2n+1}a_2^2+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n}(\Gamma_1+1)a_2^2=
[\frac{1+\lambda}{\lambda}\Gamma_2^na_3+\frac{1+\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2+\frac{1+\lambda}{2\lambda}j_2(x)b_1+j_3(x)\frac{b_1^2}{4}+\frac{j_2(x)}{2}(b_2-\frac{b_1^2}{2})]
\tag{39}\] \[\begin{gathered}
\frac{(\Gamma_2-1)(1+\lambda)}{\lambda
j_2(x)}\Gamma_2^na_3+\frac{4\lambda\Gamma_1-1-3\lambda}{2\lambda^2
j_2(x)}\Gamma_1^{2n}a_2^2=b_2-\left(\frac{j_2(x)-j_3(x)}{j_2(x)}\right)\frac{b_1^2}{2}+\frac{1+\lambda}{\lambda}b_1
\end{gathered} \tag{40}\] Comparing the coefficient of \(z\) in \(37\)
\[[\frac{1+\lambda}{2\lambda}\Gamma_1^n(\Gamma_1+1)]a_2
= \frac{1+\lambda}{\lambda}\Gamma_1^na_2+\frac{j_2(x)}{2}d_1\]
\[-[\frac{1+\lambda}{2\lambda}]\Gamma_1^n(\Gamma_1-1)a_2=
j_2(x)\frac{d_1}{2} \tag{41}\] \[-a_2=\frac{\lambda
j_2(x)d_1}{(1+\lambda)\Gamma_1^n(\Gamma_1-1)} \tag{42}\] \[\begin{gathered}
(2a_2^2-a_3)+\frac{2}{\lambda}\Gamma_1^{2n+1}a_2^2+\frac{1-\lambda}{2\lambda^2}\Gamma_1^{2n}(\Gamma_1+1)a_2^2=
[\frac{1+\lambda}{\lambda}\Gamma_2^n(2a_2^2-a_3)+\\\frac{1+\lambda}{2\lambda^2}\Gamma_1^{2n}a_2^2+\frac{1+\lambda}{2\lambda}j_2(x)d_1+j_3(x)\frac{d_1^2}{4}+\frac{j_2(x)}{2}(d_2-\frac{d_1^2}{2})]
\end{gathered} \tag{43}\] \[\begin{gathered}
\frac{(\Gamma_2-1)(1+\lambda)}{\lambda
j_2(x)}\Gamma_2^n(2a_2^2-a_3)+\frac{4\lambda\Gamma_1-1-3\lambda}{2\lambda^2
j_2(x)}\Gamma_1^{2n}a_2^2=d_2-\left(\frac{j_2(x)-j_3(x)}{j_2(x)}\right)\frac{d_1^2}{2}+\frac{1+\lambda}{\lambda}d_1
\end{gathered} \tag{44}\] Substituting \(39\) into \(43\), gives
\[-\frac{\lambda
j_2(x)b_1}{(1+\lambda)\Gamma_1^n(\Gamma_1-1)}=\frac{\lambda
j_2(x)d_1}{(1+\lambda)\Gamma_1^n(\Gamma_1-1)} \tag{45}\] \[\Longrightarrow b_1=-d_1\Rightarrow
b_1^2=d_1^2\Rightarrow b_1^3=-d_1^3 \tag{46}\] Square and add \(38\) and \(42\), we have
\[2\frac{(1+\lambda)^2}{\lambda^2}\Gamma_1^{2n}(\Gamma_1-1)^2a_2^2=
j_2^2(x)[b_1+d_1]^2 \tag{47}\] Adding \(66\) and \(45\) with further simplification
\[\begin{gathered}
\frac{2(\Gamma_2-1)(1+\lambda)}{\lambda
j_2(x)}\Gamma_2^na_2^2+\frac{4\lambda\Gamma_1-1-3\lambda}{\lambda^2
j_2(x)}\Gamma_1^{2n}a_2^2=\\(b_2+d_2)-\left(\frac{j_2(x)-j_3(x)}{j_2(x)}\right)(\frac{b_1^2}{2}+\frac{d_1^2}{2})+\frac{1+\lambda}{\lambda}(b_1+d_1)
\end{gathered} \tag{48}\] \[\begin{gathered}
\left[\frac{2(\Gamma_2-1)(1+\lambda)}{\lambda
j_2(x)}\Gamma_2^n+\frac{4\lambda\Gamma_1-1-3\lambda}{\lambda^2
j_2(x)}\Gamma_1^{2n}\right]a_2^2=\\(b_2+d_2)-\left(\frac{j_2(x)-j_3(x)}{j_2(x)}\right)(\frac{b_1^2}{2}+\frac{d_1^2}{2})+\frac{1+\lambda}{\lambda}(b_1+d_1)
\end{gathered} \tag{49}\] Using \(47\) in
\(50\), gives
\[\left[\frac{2\lambda(\Gamma_2-1)(1+\lambda)\Gamma_2^n+(4\lambda\Gamma_1-1-3\lambda)\Gamma_1^{2n}}{\lambda^2
j_2(x)}\right]a_2^2=(b_2+d_2)-\left(\frac{j_2(x)-j_3(x)}{j_2(x)}\right)(\frac{d_1^2}{2}) \tag{50}\]
Using \(47\) to simplify \(48\), gives
\[d_1^2=\frac{(1+\lambda)^2}{j_2^2(x)\lambda^2}\Gamma_1^{2n}(\Gamma_1-1)^2a_2^2 \tag{51}\]
substituting \(52\) in \(51\) with further simplification, we
obtain
\[\begin{gathered}
a_2^2=\frac{\lambda^2 j_2^3(x)(b_2+d_2)}{2\lambda
j_2^2(x)(\Gamma_2-1)(1+\lambda)\Gamma_2^n+j_2^2(x)(4\lambda\Gamma_1-1-3\lambda)\Gamma_1^{2n}+(j_2(x)-j_3(x))(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2}
\end{gathered} \tag{52}\] \[\begin{gathered}
|a_2|^2= \left|\frac{\lambda^2 j_2^3(x)(b_2+d_2)}{2\lambda
j_2^2(x)(\Gamma_2-1)(1+\lambda)\Gamma_2^n+j_2^2(x)(4\lambda\Gamma_1-1-3\lambda)\Gamma_1^{2n}+(j_2(x)-j_3(x))(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2}\right|
\end{gathered} \tag{53}\] Using Lemma \(1\) and \(2\), we have
\[|a_2|^2 = \left| \frac{4\lambda^2 |hx|^2
|hx|}{
\left| 2\lambda (hx)^2 (\Gamma_2 – 1)(1 + \lambda) \Gamma_2^n
\right. } \\
\left. + (hx)^2 (4\lambda \Gamma_1 – 1 – 3\lambda) \Gamma_1^{2n}
\right. \\
\left. + (hx – (phx^2 + qa))(1 + \lambda)^2 \Gamma_1^{2n} (\Gamma_1
– 1)^2 \right|
\right|. \tag{54}\]
Therefore,
\[|a_2| \leq
\frac{2\lambda\sqrt{|hx|}|hx|}{\sqrt{[2\lambda(1+\lambda)\Gamma_2^{n}(\Gamma_2
-1)+(4\lambda\Gamma_1-1-3\lambda)](hx)^2+(hx-(phx^2+qa))(1+\lambda)^2(\Gamma_1-1)^2\Gamma_1^{2n}}} \tag{55}\]
Subtracting \(45\) from \(46\) and using \(47\) for further simplification, we
have
\[\frac{2(\Gamma_2-1)(1+\lambda)\Gamma_2^n}{\lambda
j_2(x)}(a_3-a_2^2)= (b_2-d_2)-\frac{2(1+\lambda)}{\lambda}d_1 \tag{56}\]
\[a_3=
\frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)}-\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n}+a_2^2 \tag{57}\]
Obtain \(a_2^2\) from \(52\) and substitute it into \(58\), gives
\[a_3=
\frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)}-\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n}+\frac{j_2^2(x)d_1^2\lambda^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2} \tag{58}\]
\[|a_3|=
|\frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)}-\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n}+\frac{j_2^2(x)d_1^2\lambda^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2}| \tag{59}\]
\[|a_3|\leq
|\frac{4\lambda(b_2-d_2)j_2(x))}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)}-\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n}|+|\frac{j_2^2(x)d_1^2\lambda^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2}| \tag{60}\]
\[|a_3|\leq
\frac{2\lambda|hx|}{\Gamma_2^n(\Gamma_2-1)(1+\lambda)}+\frac{2|hx|}{(\Gamma_2-1)\Gamma_2^n}+\frac{4\lambda^2(hx)^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2} \tag{61}\]
\[|a_3|\leq
\frac{2\lambda|hx|(1+2\lambda)}{\Gamma_2^n(\Gamma_2-1)(1+\lambda)}+\frac{4\lambda^2(hx)^2}{(1+\lambda)^2\Gamma_1^{2n}(\Gamma_1-1)^2} \tag{62}\]
From \(56\) and \(63\), we have the desired result. 0◻ ◻
Remark 3. It has been observed that when \(n=0\), \(t=1\) and \(\beta=\mu\),the result obtained gives an improved estimate of the second coefficient of \(W_\Sigma(x;z)\) studied by Srivastava, et al(2018).
Theorem 2. Let \(f(z) \in I^{n+1}_{\Gamma_m,\lambda}(x,z)\), also let \(\sigma \in \Re\) then \[ |a_3-\sigma a_2^2| \leq \left\{ \begin{array}{l} \frac{4(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|},~~\text{for}~~ |1-\sigma|\leq\frac{(1+2\lambda)|hx||\Upsilon+\Psi|}{2\lambda^2|hx|^3|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}\\ \frac{8|1-\sigma|\lambda^2|hx|^3}{|\Upsilon+\Psi|},~~\text{for}~~|1-\sigma|\geq \frac{1+2\lambda|hx||\Upsilon+\Psi|}{2\lambda^2|hx|^3|\Gamma_2^n(\Gamma_2-1(1+\lambda))|} \end{array} \right. \tag{63}\]
Where
\[\Upsilon=[2\lambda(1+\lambda)\Gamma_2^{n}(\Gamma_2
-1)+(4\lambda\Gamma_1-1-3\lambda)](hx)^2 \tag{64}\] and
\[\Psi=(hx-(phx^2+qa))(1+\lambda)^2(\Gamma_1-1)^2\Gamma_1^{2n} \tag{65}\]
Proof. From \(53\) and
\(58\),
\[a_3-\sigma
a_2^2=\frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)}-\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n}+a_2^2-\sigma
a_2^2 \tag{66}\] Let
\[\Upsilon=[2\lambda(1+\lambda)\Gamma_2^{n}(\Gamma_2
-1)+(4\lambda\Gamma_1-1-3\lambda)](hx)^2\]
and
\[\Psi=(hx-(phx^2+qa))(1+\lambda)^2(\Gamma_1-1)^2\Gamma_1^{2n}\]
\[a_3 – \sigma a_2^2 =
\frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)} –
\frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n} +
(1-\sigma)\left(\frac{\lambda^2j_2^3(b_2+d_2)}{\Upsilon+\Psi}\right) \tag{67}\]
\[|a_3-\sigma a_2^2| = \left| \frac{\lambda(b_2-d_2)j_2(x)}{2\Gamma_2^n(\Gamma_2-1)(1+\lambda)} – \frac{j_2(x)d_1}{(\Gamma_2-1)\Gamma_2^n} + (1-\sigma)\left\{ \frac{\lambda^2j_2^3(b_2+d_2)}{\Upsilon+\Psi} \right\} \right| \tag{68}\]
\[|a_3-\sigma a_2^2|\leq
\frac{2(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}+|1-\sigma|\frac{4\lambda^2|hx|^3}{|\Upsilon+\Psi|} \tag{69}\]
If
\[|1-\sigma|\frac{4\lambda^2|hx|^3}{|\Upsilon+\Psi|}\leq\frac{2(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}, \tag{70}\]
then
\[|a_3-\sigma a_2^2|\leq
\frac{4(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}\ \tag{71}\]
Also, if
\[|1-\sigma|\frac{4\lambda^2|hx|^3}{|\Upsilon+\Psi|}\geq\frac{2(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}, \tag{72}\]
then
\[|a_3-\sigma a_2^2|\leq
\frac{8|1-\sigma|\lambda^2|hx|^3}{|\Upsilon+\Psi|} \tag{73}\] From \(71\),\(72\),\(73\) and \(74\), we have the desired result. 0◻ ◻
Corollary 1: If \(f(z) \in S\) and \(I^{n+1}_{\Gamma_m,\lambda}(x,z)\), then
\[|a_3-a_2^2|\leq \frac{4(1+2\lambda)|hx|}{|\Gamma_2^n(\Gamma_2-1)(1+\lambda)|}\]
The investigation in this work was driven from Geometric Function
Theory\((GFT)\) where several
interesting and various special functions and polynomials can be
studied. Horadam polynomials \(j_n(x)\)
and a generalized differential operator were used to define a new class
\(I^{n+1}_{\Gamma_m,\lambda}(x,z)\) of
bi-univalent functions by means of subordination. We obtained upper
estimates for initial coefficient and Fekete-szegö functional of the
newly defined class of bi-univalent function.
The results obtained have important roles in complex and potential
theory, with application in real life like: error analysis in numerical
method, controlled growth of analytic function, design of approximation
algorithms, electrical engineering, physics and materials science,
statistics and probability theory.In essence, these mathematical
concepts provide tools and insights that transcend pure mathematics,
finding applications in diverse fields where analytical and
computational methods are employed for modeling, analysis and
design.
The geometric properties of the function class \(I^{n+1}_{\Gamma_m,\lambda}(x,z)\) vary when
the parameters in the class are changed: when \(n=0\),\(t=1\), \(\beta=\mu\), then we have
\[\frac{1}{2}[\frac{zf'(z)}{f(z)}+(\frac{zf'(z)}{f(z)})^{\frac{1}{\lambda}}]\prec
\prod (x,z)+1-a\]
Also, when \(n=0\), \(t=1\), \(\beta=\mu\), \(a=p=x\), \(\lambda=1\) and \(q=0\), then we have \[\frac{zf'(z)}{f(z)}\prec
\frac{1+z}{1-z}\].
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