1 Introduction
The q-calculus has great applications in the space of geometric functions theory because of their usefulness in the area of ordinary fractional calculus and optimal control problems. Jackson (see [1,2]) developed the concept of qintegral and q-derivative and much later its geometrical interpretation was identified through studies of quantum groups. This has attracted the attention of several researchers. Researchers all over the globe have applied it to construct and investigate several classes of analytic and bi-univalent functions. For recent expository work on so called post-quantum calculus or (p,q) calculus, see [3,4]. We here recall the definition of fractional q-calculus operators of complex valued function f(z).
Definition 1.1. (see [3]) The (p,q)-derivative of f is defined as:
\[(D_{p,q}f)(z) = \begin{cases} \frac{f(pz) - f(qz)}{(p-q)z} \\ f'(0) \end{cases} (z \neq 0)\] (1)
provided that f is differentiable at 0. Now , = [], −1 , where
\[[n]_{p,q} = \frac{p^n - q^n}{p - q} (0 < q < p \le 1)\] (2)
refers to a twin-basic number. For p=1, the Jackson (p,q)-derivative reduces to the Jackson q-derivative given by:
\[(D_q f)(z) = \frac{f(z) - f(qz)}{(1 - q)z} (z \neq 0).\]
The class of all analytic functions f normalized by f(0) = f'(0) - 1 = 0 is given by:
\[f(z) = z + \sum_{n=2}^{\infty} a_n z^n (z \in U)\] (3)
where \(U:=\{z\in C\colon |z|<1\}\) represents the open unit disk. We denote such class by A. Let S represent the class of all analytic univalent functions of the form (3) in U. Let f, \(g\in A\). Then f is subordinate to g, written as \(f\prec g\), if there is an analytic function g in g in g in g in g in g in g in g in g in g in g in g in g in g is subordinate to g, written as g in there is an analytic function g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g in g
\[g(w) = f^{-1}(w)\]
= \(w - a_2 w^2 + (2a_2^2 - a_3)w^3 - (5a_2^3 - 5a_2a_3 + a_4)w^4 + \cdots (4)\)
If both \(f, f^{-1} \in S\) then f is said to be bi-univalent in U. The class of all functions f given by (3) is denoted by \(\Sigma\). For a detailed history and other related properties of functions in the class \(\Sigma\), see recent works in [8-13].
For a function f given by (3), a simple calculation shows that
\[D_{p,q}f(z) = 1 + \sum_{n=2}^{\infty} [n]_{p,q} a_n z^{n-1}.\] (5)
The (p,q)-analogue of Salagean differential operator \(R_{p,q}^k: A \to A(k \in N_0 = N \cup \{0\})\) is defined by:
\[R_{p,q}^{0}f(z) = f(z)\] \[R_{p,q}^{1}f(z) = z\left(D_{p,q}f(z)\right),\] ... \[R_{p,q}^{k}f(z) = R_{p,q}^{1}(R_{p,q}^{k-1}f(z))\] (6)
Thus, for a function f(z) of the form (3), we have:
\[R_{p,q}^{k}f(z) = z + \sum_{n=2}^{\infty} [n]_{p,q}^{k} a_{n}z^{n}.\] (7)
Similarly, for a function g of the form (4), we have:
\[R_{p,q}^{k}g(w) = w - [2]_{p,q}^{k}a_{2}w^{2} + (2a_{2}^{2} - a_{3})[3]_{p,q}^{k}w^{3} - (5a_{2}^{3} - 5a_{2}a_{3} + a_{4})[4]_{p,q}^{k}w^{4} + \cdots\] (8)
From above, we observe that:
\[\lim_{p \to 1, q \to 1^{-}} R_{p,q}^{k} f(z) = z + \lim_{p \to 1, q \to 1^{-}} \sum_{n=2}^{\infty} [n]_{p,q}^{k} a_{n} z^{n}\] \[= z + \sum_{n=2}^{\infty} n^{k} a_{n} z^{n} = D^{k} f(z), \tag{9}\] where \(D^k\) is the Salagean differential operator which was defined in [14] and has been studied by several authors.
Chebyshev polynomials of the first and second kind and their properties have been studied by several researchers (see, for details [15,16]). We consider
\[L(z,t) = \frac{1}{1-2tz+z^2}(z \in U)\] as its generating function. Taking \(t = \cos \alpha\), \(\alpha \in \left(-\frac{\pi}{3}, \frac{\pi}{3}\right)\), we have:
\[L(z,t) = \frac{1}{1 - 2\cos\alpha z + z^2}\] \[= 1 + 2\cos\alpha z + (3\cos^2\alpha - \sin^2\alpha)z^2 + \cdots\] \[= 1 + U_1(t)z + U_2(t)z^2 + \cdots \quad (z \in U, t \in (-1,1)),\] (10)
where \(U_{n-1}(t) = \frac{\sin(\cos^{-1}t)}{\sqrt{1-t^2}} (n \in \mathbb{N})\). Thus we have
\[U_1(t) = 2t,\] \(U_3(t) = 8t^3 - 4t,\) \(U_2(t) = 4t^2 - 1,\) \(U_4(t) = 16t^4 - 12t^2 + 1, ...\) (11)
Recently, several researchers, Altinkaya and Yalcin [17-19], Bulut et al. [20,21] Guney et al. [22] and Caglar [23] (also see [24]) to mention a few, have obtained Fekete-Szego inequalities and some coefficient bounds for different subclasses of bi-univalent functions. Motivated by the above researchers, we consider two subclasses of bi-univalent functions that are obtained by using the \(D_{p,q}\) operator of the Salagean type associated with the Chebyshev polynomial.
Definition 1.2. A function \(f \in \Sigma\) defined as Eq. (3) belongs to the function class \(R_{\Sigma,p,q}^k(\gamma,t)(0 \le \gamma \le 1)\) if the conditions
\[(1 - \gamma) \frac{R_{p,q}^{k+1} f(z)}{R_{p,q}^{k} f(z)} + \gamma \frac{R_{p,q}^{k+2} f(z)}{R_{p,q}^{k+1} f(z)} < L(z,t) \left(\frac{1}{2} < t < 1; z \in U\right), \tag{12}\] and
\[(1 - \gamma) \frac{R_{p,q}^{k+1}g(w)}{R_{n,q}^{k}g(w)} + \gamma \frac{R_{p,q}^{k+2}g(w)}{R_{n,q}^{k+1}g(w)} < L(w,t) \left(\frac{1}{2} < t < 1; w \in U\right), \tag{13}\] are satisfied, where g is stated in (4).
By specializing the parameters \(\gamma\), p, q and k in the above definition, we obtain the various subclasses of \(\Sigma\).
Definition 1.3. A function \(f \in \Sigma\) belongs to the function class \(T_{\Sigma,p,q}^k(\beta,t)\) if
\[(1 - \beta) \frac{R_{p,q}^k f(z)}{z} + \beta (R_{p,q}^k f(z))' < L(z, t), \tag{14}\] and
\[(1 - \beta) \frac{R_{p,q}^k g(w)}{w} + \beta (R_{p,q}^k g(w))' < L(w,t)\] (15)
\((0 \le \beta \le 1, \frac{1}{2} < t < 1; z, w \in U)\), hold where \(R_{p,q}^k f(z)\) and \(R_{p,q}^k g(w)\) are given by Eq. (7) and Eq. (8) respectively.
Remark 1.4. For \(p \to 1, q \to 1^-\), we get the class \(T_{\Sigma,1,1^-}^k(\beta,t) = F_{\Sigma}^k(\beta,L(z,t))\) consists of function \(f \in \Sigma\) and satisfying
\[(1-\beta)^{\frac{D^k f(z)}{z}} + \beta (D^k f(z))' < L(z,t)\] and
\[(1-\beta)^{\frac{D^kg(w)}{w}} + \beta(D^kg(w))' < L(w,t).\]
This class is due to Guney et al.[22].
Remark 1.5. For \(p \to 1\), \(q \to 1^-\) and k=0, we obtain the class \(T^0_{\Sigma,1,1^-}(\beta,t) = B_{\Sigma}(\beta,t)\) (see[20, 21]) where \(f \in \Sigma\) satisfying
\[(1-\beta)\frac{f(z)}{z} + \beta(f(z))' < L(z,t)\]
\[(1-\beta)\frac{g(w)}{w} + \beta(g(w))' \prec L(w,t).\]
In this work, we investigate the first two coefficient bounds and Fekte-Szego inequalities in the above newly constructed function classes by using the Chebyshev polynomial.
2 Coefficient Bounds
In the following theorems, we establish Chebyshev polynomial bounds \(\left|a_{2}\right|\) and \(|a_3|\) for the function classes \(R_{\sum p,q}^k(\gamma,t)\) and \(T_{\sum p,q}^k(\beta,t)\).
Theorem 2.1. Assume that \(f \in \Sigma\) defined as Eq. (3) is in the class \(R_{\Sigma,p,q}^k(\gamma,t) \left(\frac{1}{2} < t < 1\right)\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(A_1 - A_2)4t^2 + A_2|}},\tag{16}\] and
\[|a_{3}| \leq \frac{4t^{2}}{[2]_{p,q}^{2k}([2]_{p,q}-1)^{2}(1+\gamma([2]_{p,q}-1))^{2}} + \frac{2t}{[3]_{p,q}^{k}([3]_{p,q}-1)(1+\gamma([3]_{p,q}-1))'}\] (17)
where
\[A_{1} = [3]_{p,q}^{k} (1 + \gamma([3]_{p,q} - 1))([3]_{p,q} - 1) - [2]_{p,q}^{2k} (1 + \gamma([2]_{p,q}^{2} - 1))([2]_{p,q} - 1),\] (18)
and
\[A_2 = [2]_{p,q}^{2k} (1 + \gamma([2]_{p,q} - 1))^2 ([2]_{p,q} - 1)^2.\] (19)
Proof: Assume that \(f \in R^k_{\Sigma,p,q}(\gamma,t)\). Definition 1.2 yields:
\[(1-\gamma)\frac{R_{p,q}^{k+1}f(z)}{R_{p,q}^{k}f(z)} + \gamma\frac{R_{p,q}^{k+2}f(z)}{R_{p,q}^{k+1}f(z)}1 + U_{1}(t)r(z) + U_{2}(t)r^{2}(z) + \cdots \qquad (20)\] and
\[(1-\gamma)\frac{R_{p,q}^{k+1}g(w)}{R_{p,q}^{k}g(w)} + \gamma\frac{R_{p,q}^{k+2}g(w)}{R_{p,q}^{k+1}g(w)} = 1 + U_1(t)s(w) + U_2(t)s^2(w) + \cdots (21)\] where r(z) and s(w) are analytic functions given by
\[r(z) = c_1 z + c_2 z^2 + c_3 z^3 + \cdots,\] \[s(w) = d_1 w + d_2 w^2 + d_3 w^3 + \cdots,\] (22)
\[s(w) = d_1 w + d_2 w^2 + d_3 w^3 + \cdots, (23)\] where r(0) = s(0) = 0, |r(z)| < 1, |s(w)| < 1 \((z, w \in U)\). If |r(z)| < 1 and |s(w)| < 1, then
\[|c_i| \le 1 \text{ and } |d_i| < 1 \text{ for all } i \in N.\] (24)
Making use of Eq. (22) in Eq. (20) and Eq. (23) in Eq. (21), we get
\[(1 - \gamma) \frac{R_{p,q}^{k+1}f(z)}{R_{p,q}^{k}f(z)} + \gamma \frac{R_{p,q}^{k+2}f(z)}{R_{p,q}^{k+1}f(z)}\] \[= 1 + U_{1}(t)c_{1}z + [U_{1}(t)c_{2} + U_{2}(t)c_{1}^{2}]z^{2} + \cdots\] (25)
and
\[(1 - \gamma) \frac{R_{p,q}^{k+1}g(w)}{R_{p,q}^{k}g(w)} + \gamma \frac{R_{p,q}^{k+2}g(w)}{R_{p,q}^{k+1}g(w)}\] \[= 1 + U_{1}(t)d_{1}w + [U_{1}(t)d_{2} + U_{2}(t)d_{1}^{2}]w^{2} + \cdots\] (26)
It follows from Eq. (7) and Eq. (8) that
\[\begin{split} &(1-\gamma)\frac{R_{p,q}^{k+1}f(z)}{R_{p,q}^{k}f(z)} + \gamma\frac{R_{p,q}^{k+2}f(z)}{R_{p,q}^{k+1}f(z)} \\ &= 1 + [2]_{p,q}^{k}(1+\gamma([2]_{p,q}-1))([2]_{p,q}-1)a_{2}z + ([3]_{p,q}-1)\{[3]_{p,q}^{k}(1+\gamma([3]_{p,q}-1))a_{3}-([2]_{p,q}-1)[2]_{p,q}^{k}(1+\gamma([2]_{p,q}^{2}-1))a_{2}^{2}\}z^{2} + \cdots \end{split} \tag{27}\] and
\[(1-\gamma)\frac{R_{p,q}^{k+1}g(w)}{R_{p,q}^{k}g(w)} + \gamma\frac{R_{p,q}^{k+2}g(w)}{R_{p,q}^{k+1}g(w)} = 1 - [2]_{p,q}^{k}(1+\gamma([2]_{p,q} - 1))([2]_{p,q} - 1)a_{2}w + [\{2([3]_{p,q} - 1)[3]_{p,q}^{k}(1+\gamma([3]_{p,q} - 1)) - ([2]_{p,q} - 1)[2]_{p,q}^{k}(1+\gamma([2]_{p,q}^{2} - 1))\}a_{2}^{2} - [3]_{p,q}^{k}(1+\gamma([3]_{p,q} - 1))([3]_{p,q} - 1)a_{3}]w^{2} + \cdots\] \[(28)\]
Using Eq. (27) in Eq. (25) and Eq. (28) in Eq. (26), we obtain:
\[1 + [2]_{p,q}^{k} \left(1 + \gamma([2]_{p,q} - 1)\right) ([2]_{p,q} - 1) a_{2}z + \left[ ([3]_{p,q} - 1)[3]_{p,q}^{k} \left(1 + \gamma([3]_{p,q} - 1)\right) a_{3} - ([2]_{p,q} - 1)[2]_{p,q}^{2k} \left(1 + \gamma([2]_{p,q}^{2} - 1)\right) a_{2}^{2} \right] z^{2} + \dots = 1 + U_{1}(t)c_{1}z + [U_{1}(t)c_{2} + U_{2}(t)c_{1}^{2}]z^{2} + \dots\] \[(29)\]
\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[= 1 + U_{1}(t)d_{1}\omega + \left[U_{1}(t)d_{2} + U_{2}(t)d_{1}^{2}\right]\omega^{2} + \cdots\] (30)
Equating the coefficients in Eq. (29) and Eq. (30), we get:
\[([2]_{p,q} - 1)[2]_{p,q}^{k} \left(1 + \gamma([2]_{p,q} - 1)\right) a_2 = U_1(t)c_1, \tag{31}\] \[-([2]_{p,q} - 1)[2]_{p,q}^{2k} \left(1 + \gamma([2]_{p,q}^2 - 1)\right) a_2^2 + ([3]_{p,q} - 1)[3]_{p,q}^{k} \left(1 + \gamma([3]_{p,q} - 1)\right) a_3 = U_1(t)c_2 + U_2(t)c_1^2, \tag{32}\] and
\[-([2]_{p,q}-1)[2]_{p,q}^{k}\left(1+\gamma([2]_{p,q}-1)\right)a_{2}=U_{1}(t)d_{1},\tag{33}\] and
\[\left\{2([3]_{p,q}-1)[3]_{p,q}^{k}\left(1+\gamma([3]_{p,q}-1)\right)-([2]_{p,q}-1)[2]_{p,q}^{2k}\left(1+\gamma([2]_{p,q}^{2}-1)\right)\right\}a_{2}^{2}-([3]_{p,q}-1)[3]_{p,q}^{k}\left(1+\gamma([3]_{p,q}-1)\right)a_{3}=U_{1}(t)d_{2}+U_{2}(t)d_{1}^{2}.\] (34)
From Eq. (31) and Eq. (33), we obtain:
\[c_1 = -d_1, (35)\] and
\[2([2]_{p,q} - 1)^{2}[2]_{p,q}^{2k} (1 + \gamma([2]_{p,q} - 1))^{2} a_{2}^{2}\] \[= U_{1}^{2}(t)(c_{1}^{2} + d_{1}^{2}).\] (36)
Adding Eq. (32) and Eq. (34) and using Eq. (36) in the resulting equation, we obtain:
\[\left[2([3]_{p,q}-1)[3]_{p,q}^{k}\left(1+\gamma([3]_{p,q}-1)\right)-2([2]_{p,q}-1)[2]_{p,q}^{2k}\left(1+\gamma([2]_{p,q}^{2}-1)\right)-\frac{U_{2}(t)}{U_{1}^{2}(t)}2[2]_{p,q}^{2k}\left([2]_{p,q}-1\right)^{2}\left[1+\gamma([2]_{p,q}-1)\right]^{2}\right]a_{2}^{2}=U_{1}(t)(c_{2}+d_{2}),\] (37)
which gives:
\[a_2^2 = \frac{(c_2 + d_2)U_1^3(t)}{2[A_1U_1^2(t) - A_2U_2(t)]},\tag{38}\] where \(A_1\) and \(A_2\) are given in Eq. (18) and Eq. (19) respectively. Applying Eq. (24) to the coefficients \(c_2\) and \(d_2\) and using Eq. (11) in Eq. (38), we get the desire estimate for \(|a_2|\).
Subtracting Eq. (34) from Eq. (32) and using Eq. (35) and Eq. (36) in the resulting equation yields:
\[a_3 = \frac{(c_1^2 + d_1^2)U_1^2(t)}{2[2]_{p,q}^{2k}([2]_{p,q} - 1)^2[1 + \gamma([2]_{p,q} - 1)]^2} + \frac{(c_2 - d_2)U_1(t)}{2[3]_{p,q}^k([3]_{p,q} - 1)[1 + \gamma([3]_{p,q} - 1)]} . (39)\]
Taking the coefficient inequalities for \(c_1\), \(c_2\), \(d_1\) and \(d_2\) from Eq. (24) and making use of Eq. (11) in Eq. (39) we get the estimate for \(|a_3|\) as stated in Eq. (17). This proves the Theorem 2.1.
Letting \(p \to 1\) and \(q \to 1^-\) in Theorem 2.1, we get the result for the class \(R_{\Sigma,1,1^{-1}}^k(\gamma,t) \equiv M_{\Sigma}^k(\gamma,L(z,t))\) due to Guney et al. [22] as follows:
Corollary 2.2 (see [22]): Let \(f \in M_{\Sigma}^{k}(\gamma, L(z, t))\). Then
\[|a_2| \leq \frac{2t\sqrt{2t}}{\sqrt{|[2(1+2\gamma)3^k - (\gamma(\gamma+5)+2)2^{2k}]4t^2 + 2^{2k}(1+\gamma)^2|}}\] and
\[|a_3| \le \frac{4t^2}{(1+\gamma)^2 2^{2k}} + \frac{t}{(1+2\gamma)3^k}\].
Letting \(\gamma = 0\) in Theorem 2.1, the following result for the function class \(R_{\Sigma,p,q}^k(0,t) \equiv N_{\Sigma,p,q}^k(t)\) is obtained.
Corollary 2.3. If \(f \in N_{\Sigma,p,q}^k(t)\), then
\[|a_2| \leq \frac{2t\sqrt{2t}}{\sqrt{|[([3]_{p,q}-1)[3]_{p,q}^k - ([2]_{p,q}-1)[2]_{p,q}^{2k+1}]4t^2 + ([2]_{p,q}-1)^2[2]_{p,q}^{2k}}}\] and
\[|a_3| \le \frac{2t}{[3]_{n,q}^k([3]_{p,q}-1)} + \frac{4t^2}{[2]_{n,q}^{2k}([2]_{p,q}-1)^2}.\]
Letting \(p \to 1\) and \(q \to 1^-\) in the above corollary, we get the following result for the class \(R_{\Sigma,1,1}^k(0,t) \equiv N_{\Sigma}^k(t)\).
Corollary 2.4. Let \(f \in N_{\Sigma}^{k}(t)\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(3^k - 2^{2k})8t^2 + 2^{2k}|}}\] and
\[|a_3| \le \frac{t}{3^k} + \frac{4t^2}{2^{2k}}.\]
Putting \(\gamma=1\) in Theorem 2.1, the result for the class \(R^k_{\Sigma,p,q}(1,t)\equiv U^k_{\Sigma,p,q}(t)\) is as follows:
Corollary 2.5. Let \(f \in U^k_{\Sigma,p,q}(t)\). Then
\[|a_2| \leq \frac{2t\sqrt{2t}}{\sqrt{|[([3]_{p,q}-1)[3]_{p,q}^{k+1}-([2]_{p,q}-1)[2]_{p,q}^{2k+3}]4t^2+[2]_{p,q}^{2k+2}([2]_{p,q}-1)^2|}},\] and
\[|a_3| \leq \tfrac{4t^2}{[2]_{p,q}^{2k+2}([2]_{p,q}-1)^2} + \tfrac{2t}{([3]_{p,q}-1)[3]_{p,q}^{k+1}}.\]
Taking \(p \to 1, q \to 1^-\) and \(\gamma = 1\) in the above theorem, the result for the class \(R_{\Sigma,1,1^-}^k(1,t) \equiv K_{\Sigma}^k(L(z,t))\) is obtained.
Corollary 2.6 (see [22]): If \(f \in K^k_{\Sigma}(L(z,t))\), then
\[|a_2| \le \frac{t\sqrt{2t}}{\sqrt{|[3^{k+1}-2^{2(k+1)}]2t^2+2^{2k}|}},\] and
\[|a_3| \le \frac{t^2}{2^{2k}} + \frac{t}{3^{k+1}}\]
Theorem 2.1 for k=0 gives
Corollary 2.7. Let the function \(f \in V_{\sum p,q}(\gamma,t) (\equiv R_{\sum p,q}^0(\gamma,t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|4M_1t^2 + M_2|'}}\]
\[|a_3| \leq \frac{4t^2}{(1+\gamma([2]_{p,q}-1))^2([2]_{p,q}-1)^2} + \frac{2t}{(1+\gamma([3]_{p,q}-1))([3]_{p,q}-1)},\] where
\[\begin{split} M_1 &= (1 + \gamma([3]_{p,q} - 1))([3]_{p,q} - 1) - ([2]_{p,q} - 1)\{1 + \gamma([2]_{p,q}^2 - 1) + ([2]_{p,q} - 1)(1 + \gamma([2]_{p,q} - 1))^2\}, \end{split}\] and
\[M_2 = (1 + \gamma([2]_{p,q} - 1))^2([2]_{p,q} - 1)^2.\]
Letting \(p \to 1\) and \(q \to 1^-\)in the above result, we get
Corollary 2.8. Let \(f \in V_{\Sigma}(\gamma, t) (\equiv R_{\Sigma, 1, 1}^{0}(\gamma, t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(1+\gamma)^2 - (\gamma+\gamma^2)4t^2|}},\] and
\[|a_3| \le \frac{4t^2}{(1+\gamma)^2} + \frac{t}{1+2\gamma}.\]
Putting \(\gamma = 0\) in the above corollary gives the following.
Corollary 2.9. Let \(f \in V_{\Sigma}(t) (\equiv V_{\Sigma}(0,t)\). Then
\[|a_2| \le 2t\sqrt{2t},\] and
\[|a_3| \le t + 4t^2.\]
Putting \(\gamma = 1\) in Corollary 2.8 gives:
Corollary 2.10. Let \(f \in Q_{\Sigma}(t) (\equiv V_{\Sigma}(1,t))\). For \(t \neq \frac{1}{\sqrt{2}}\), we have
\[|a_2| \le \frac{t\sqrt{2t}}{\sqrt{|1-2t^2|}}\] and
\[|a_3| \le t^2 + \frac{t}{3}.\]
Theorem 2.11. Let \(f \in T^k_{\Sigma,p,q}(\beta,t)\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|[[3]_{p,q}^k(1+2\beta)-[2]_{p,q}^k(1+\beta)^2]4t^2+[2]_{p,q}^{2k}(1+\beta)^2|}},\tag{40}\]
\[|a_3| \le \frac{2t}{(1+2\beta)[3]_{p,q}^k} + \frac{4t^2}{(1+\beta)^2[2]_{p,q}^{2k}}.\] (41)
Proof: Let \(f \in T^k_{\Sigma,p,q}(\beta,t)\). Proceeding as before, we have
\[(1+\beta)[2]_{p,q}^{k}a_{2} = U_{1}(t)c_{1}, \tag{42}\]
\[(1+2\beta)[3]_{n,q}^{k}a_3 = U_1(t)c_2 + U_2(t)c_1^2 \tag{43}\] and
\[-(1+\beta)[2]_{n,q}^{k}a_2 = U_1(t)d_1, \tag{44}\]
\[(1+2\beta)(2a_2^2-a_3)[3]_{p,q}^k = U_1(t)d_2 + U_2(t)d_1^2, \tag{45}\]
It follows from Eq. (42) and Eq. (44) that
\[c_1 = -d_1, (46)\]
\[2(1+\beta)^{2}[2]_{p,q}^{2k}a_{2}^{2} = U_{1}^{2}(t)(c_{1}^{2}+d_{1}^{2}). \tag{47}\]
Similarly, from Eq. (43) and Eq. (45) we have:
\[2(1+2\beta)[3]_{p,q}^{k}a_{2}^{2} = U_{1}(t)(c_{2}+d_{2}) + U_{2}(t)(c_{1}^{2}+d_{1}^{2}). \tag{48}\]
Using Eq. (47) in Eq. (48) and simplifying we get:
\[a_2^2 = \frac{(c_2 + d_2)U_1^2(t)}{2[(1 + 2\beta)[3]_{p,q}^k U_1^2(t) - (1 + \beta)^2[2]_{p,q}^{2k} U_2(t)]}.\] (49)
Putting the values of \(U_1(t)\), \(U_2(t)\) from Eq. (11) and using Eq. (24) in Eq. (49) we get the desire estimate for \(|a_2|\) as given by Eq. (40).
Subtracting Eq. (45) from Eq. (43) and making use of Eq. (46) and Eq. (47) in the resulting equation and simplifying, we get:
\[a_3 = \frac{(c_2 - d_2)U_1(t)}{2(1 + 2\beta)[3]_{n,q}^k} + \frac{(c_2^2 - d_2^2)U_1^2(t)}{2(1 + 2\beta)^2[3]_{n,q}^{2k}}\] (50)
Using Eq. (11) and Eq. (24) in Eq. (50) we get the bounds for \(|a_3|\). The proof of Theorem 2.11 is completed.
Taking \(q \to 1^-\), \(p \to 1\) in Theorem 2.11, the result for the class \(T^k_{\Sigma,1,1^-}(\beta,t) (\equiv F^k_{\Sigma}(\beta,L(z,t)))\) is obtained.
Corollary 2.12. Let \(f \in \Sigma\) given by Eq. (3) be in the class \(F_{\Sigma}^{k}(\beta, L(z, t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|[(1+2\beta)3^k - (1+\beta)^2 2^{2k}]4t^2 + (1+\beta)^2 2^{2k}]'}}\] and
\[|a_3| \le \frac{2t}{(1+2\beta)3^k} + \frac{4t^2}{(1+\beta)^2 2^{2k}}.\]
Putting \(\beta=0\) in Corollary 2.12 we get the result for the function class \(T^k_{\Sigma,1,1^-}(0,t)\equiv F^k_\Sigma(L(z,t))\) as follows:
Corollary 2.13 (see [22]): Let \(f \in F_{\Sigma}^k(L(z,t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|[3^k - 2^{2k}]4t^2 + 2^{2k}]'}}\] and
\[|a_3| \le \frac{2t}{3^k} + \frac{4t^2}{2^{2k}} \ .\]
Corollary 2.13 for \(\beta = 1\) yields the result for the class \(T_{\sum,1,1}^k(1,t) \equiv H_{\sum}^k(L(z,t))\) as below.
Corollary 2.14. Let \(f \in H^k_{\Sigma}(L(z,t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|[3^{k+1}-2^{2(k+1)}]4t^2+2^{2(k+1)}|}},\] and
\[|a_3| \le \frac{2t}{3^{k+1}} + \frac{t^2}{2^{2k}}.\]
Corollary 2.12 for k=0 gives the result for the class \(T^0_{\Sigma,1,1^-}(\beta,t) \equiv F_{\Sigma}(\beta,L(z,t))\) as below.
Corollary 2. 15. Let \(f \in F_{\Sigma}(\beta, L(z, t))\). Then
\[|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(\beta+1)^2 - 4\beta^2 t^2|}}\]
\[|a_3| \le \frac{2t}{1+2\beta} + \frac{4t^2}{(1+\beta)^2}.\]
Putting \(\beta=0\) in Corollary 2.15 gives the result for the function class \(T^0_{\Sigma,1,1^-}(0,t)\equiv T_\Sigma(t)\).
Corollary 2. 16. Let \(f \in T_{\Sigma}(t)\). Then
\[|a_2| \leq 2t\sqrt{2t}\] and
\[|a_3| \leq 2t + 4t^2\].
Letting \(\beta = 1\) in Corollary 2.15, we get the result for the class \(F_{\Sigma}(1, L(z, t)) \equiv F_{\Sigma}(L(z, t))\).
Corollary 2.17. Let \(f \in F_{\Sigma}(L(z,t))\). Then
\[|a_2| \le \frac{t\sqrt{2t}}{\sqrt{1-t^2}},\] and
\[|a_3| \le t^2 + \frac{2}{3}t .\]
3 Fekete-Szego Inequalities
In the following section, we obtain the Fekete-Szego problems for the function class \(R_{\Sigma,p,q}^k(\gamma,t)\) and \(T_{\Sigma,p,q}^k(\beta,t)\) as follows:
Theorem 3.1. Let \(f \in R^k_{\Sigma,p,q}(\gamma,t)\). Then
\[\left|a_{3}-\eta a_{2}^{2}\right| \leq \begin{cases} \frac{2t}{([3]_{p,q}-1)[3]_{p,q}^{k}(1+\gamma\left([3]_{p,q}-1\right)\right)} & \left|\eta-1\right| \leq \left|\frac{\frac{A_{2}}{4t^{2}}+M_{3}}{M_{5}}\right| \\ \frac{8t^{3}\left|1-\eta\right|}{\left|\left|M_{3}\right|4t^{2}+A_{2}\right|} & \left|\eta-1\right| \geq \left|\frac{\frac{A_{4}}{4t^{2}}+M_{3}}{M_{5}}\right|, \end{cases}\](51)
where
\[M_3 = M_5 - M_4 - A_2, (52)\]
\[M_4 = ([2]_{p,q} - 1)[2]_{p,q}^{2k} (1 + \gamma([2]_{p,q}^2 - 1))\] (53)
\[M_5 = ([3]_{p,q} - 1)[3]_{p,q}^k (1 + \gamma([3]_{p,q} - 1))\] (54)
and \(A_2\) is defined in Eq. (19).
Proof: It follows from Eq. (32) and Eq. (34) that
\[a_{3} - \eta a_{2}^{2} = (1 - \eta) \frac{(c_{2} + d_{2})U_{1}^{2}(t)}{2[(M_{5} - M_{4})U_{1}^{2}(t) - A_{2}U_{2}(t)]} + \frac{(c_{2} - d_{2})U_{1}(t)}{2M_{5}}\] \[= U_{1}(t) \left[ \left( g(\eta) + \frac{1}{2M_{5}} \right) c_{2} + \left( g(\eta) - \frac{1}{2M_{5}} \right) d_{2} \right], \tag{55}\] where
\[g(\eta) = \frac{(1-\eta)U_1^2(t)}{2[(M_5 - M_4)U_1^2(t) - A_2U_2(t)]}.\] (56)
Taking the values of \(U_1(t)\) and \(U_2(t)\) from Eq. (11) and substituting it in Eq. (56) we conclude that
\[\left| a_{3} - \eta a_{2}^{2} \right| \leq \begin{cases} \frac{2t}{M_{5}} & 0 \leq |g(\eta)| \leq \frac{1}{2M_{5}} \\ 4t|g(\eta)| & |g(\eta)| \geq \frac{1}{2M_{5}}. \end{cases}\] (57)
The estimate Eq. (51) follows from Eq. (57). The proof of Theorem 3.1 is thus completed.
Taking \(p \to 1\) and \(q \to 1^-\) in Theorem 3.1 yields:
Corollary 3.2. Let \(f \in M^k_{\Sigma}(\gamma, L(z,t)) (\equiv R^k_{\Sigma,1,1^-}(\gamma,t))\). Then
\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]
Theorem 3.1 for \(\eta = 1\) gives the following:
Corollary 3.3. Let \(f \in R^k_{\Sigma,p,q}(\gamma,t)\) . We have
\[|a_3 - a_2^2| \le \frac{2t}{M_r}\]
Letting \(\eta = 1\) in Corollary 3.2 we have:
Corollary 3.4 (see [22]): Let \(f \in M_{\Sigma}^k(\gamma, L(z, t))\). We have
\[|a_3 - a_2^2| \le \frac{t}{(1+2\nu)3^k}\]
Theorem 3.1 for \(\gamma = 0\) gives
Corollary 3.5. Let \(f \in N_{\Sigma,p,q}^k(t) (\equiv R_{\Sigma,p,q}^k(0,t)\). Then
\[\left|a_3 - \eta a_2^2\right| \le\]
\[\begin{cases} \frac{2t}{[3]_{p,q}^{k}([3]_{p,q}-1)}, |\eta-1| \leq \frac{\left|\frac{[2]_{p,q}^{2k}([2]_{p,q}-1)^{2}}{4t^{2}} + [3]_{p,q}^{k}([3]_{p,q}-1) - [2]_{p,q}^{2k+1}([2]_{p,q}-1)\right|}{[3]_{p,q}^{k}([3]_{p,q}-1)} \\ \frac{8t^{3}|1-\eta|}{\left|[[3]_{p,q}^{k}([3]_{p,q}-1) - [2]_{p,q}^{2k+1}([2]_{p,q}-1)]4t^{2} + [2]_{p,q}^{2k}([2]_{p,q}-1)^{2}\right|}, \\ |\eta-1| \geq \frac{\left|\frac{[2]_{p,q}^{2k}([2]_{p,q}-1)^{2}}{4t^{2}} + [3]_{p,q}^{k}([3]_{p,q}-1) - [2]_{p,q}^{2k+1}([2]_{p,q}-1)\right|}{[3]_{p,q}^{k}([3]_{p,q}-1)} \end{cases}\]
Taking \(p \to 1\) and \(q \to 1^-\) in Corollary 3.5, the result for the class \(N_{\Sigma}^k(t) \equiv N_{\Sigma,1,1^-}^k(t)\) is obtained.
Corollary 3.6. Let \(f \in N_{\Sigma}^{k}(t)\). Then for any real number \(\eta\),
\[|a_3 - \eta a_2^2| \le \begin{cases} \frac{t}{3^k} & |\eta - 1| \le |\frac{\frac{2^{2k}}{8t^2} + 3^k - 2^{2k}}{3^k}| \\ \frac{8t^3|1 - \eta|}{|(3^k - 2^{2k})8t^2 + 2^{2k}|} & |\eta - 1| \ge |\frac{\frac{2^{2k}}{8t^2} + 3^k - 2^{2k}}{3^k}| \end{cases}\]
Taking \(\eta = 1\) and k = 0 in Corollary 3.6 we get the estimate for the class \(N_{\Sigma}(t) \equiv N_{\Sigma}^{0}(t)\).
Corollary 3.7. Let \(f \in \Sigma\) given by Eq. (3) be in the class \(N_{\Sigma}(t)\). Then \(|a_3 - a_2^2| \le t\).
Theorem 3.8. Let \(f \in \Sigma\) given by Eq. (3) be in the class \(T_{\Sigma,p,q}^k(\beta,t)\). Then for any \(\eta \in R\), we have:
\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]
Proof: From Eq. (43) and Eq. (45) we have:
\[a_{3} - \eta a_{2}^{2} = (1 - \eta) \frac{U_{1}^{3}(t)(c_{2} + d_{2})}{2[(1 + 2\beta)[3]_{p,q}^{k} U_{1}^{2}(t) - (1 + \beta)^{2}[2]_{p,q}^{2k} U_{2}(t)]} + \frac{U_{1}(t)(c_{2} - d_{2})}{2(1 + 2\beta)[3]_{p,q}^{k}}\] \[= U_{1}(t) \left\{ \left[ s(\eta) + \frac{1}{2(1 + 2\beta)[3]_{p,q}^{k}} \right] c_{2} + \left[ s(\eta) - \frac{1}{2(1 + 2\beta)[3]_{p,q}^{k}} \right] d_{2} \right\}, \quad (58)\] where
\[s(\eta) = \frac{(1-\eta)U_1^2(t)}{2[(1+2\beta)[3]_{p,q}^k U_1^2(t) - (1+\beta)^2[2]_{p,q}^{2k} U_2(t)]}.\] (59)
In view of (11), we obtain:
\[\left|a_{3} - \eta a_{2}^{2}\right| \leq \begin{cases} \frac{2t}{(1+2\beta)[3]_{p,q}^{k}} & 0 \leq |s(\eta)| \leq \frac{1}{2(1+2\beta)[3]_{p,q}^{k}} \\ 4t|s(\eta)| & |s(\eta)| \geq \frac{1}{2(1+2\beta)[3]_{p,q}^{k}}. \end{cases}\](60)
The estimates of Theorem 3.8 follow from Eq. (60). This completes the proof.
Remark 3.9. Many corollaries will be generated by varying parameters involved in Theorem 3.8.
4 Conclusion
A good amount of literature is available for the first few coefficients and the Fekete-Szego problem for different subclasses of univalent and bi-univalent analytic functions by making use of the class of Caratheodory functions. In the present investigation, the authors have introduced newly constructed bi-univalent analytic function classes \(R_{\Sigma,p,q}^k(\gamma,t)\) and \(T_{\Sigma,p,q}^k(\beta,t)\) associated with the Chebyshev polynomials by using the Salagean (p,q)-differential operator and obtained initial coefficients and Fekete-Szego problems for the above mentioned classes. The generalization of some of the previous results studied by various researchers was obtained. The sigmoid function and Faber polynomial can be used to derive similar results for the classes studied.
Acknowledgement
The authors express their thanks to the editor and anonymous referees for their comments and suggestions to improve the contents. Further, the present investigation of the second-named author was supported by CSIR research project scheme no. 25(0278)/17/EMR-II, New Delhi, India.