Certain novel generalized subclasses of uniformly starlike and convex functions: coefficient characterizations and inclusion properties through Bessel and Gaussian hypergeometric functions

Lemma 2.1. [42] If $p, a, c \in \mathbb{C}$ and $m \neq 0, -1, -2, ...$ , then the function $v_p$ satisfies the recursive relation $$4m\upsilon_p'(z)=-c\upsilon_{p+1}(z),\quad \forall z\in\mathbb{U}.$$

Lemma 2.2. [27] If $f \in \Re^{\tau}(A, C)$ has the functions of the form (2.1), then $$|a_n| \le (A-C)\frac{|\tau|}{n}, \quad n \in \mathbb{N} \{0\}.$$ In the ensuing section, we initially derive the characterization properties, specifically related to the coefficient bounds, of the newly established subclasses of analytic functions $M(\tau, \rho, \nu)$ and $N(\tau, \rho, \nu)$ as expressed in Theorems 3.1 and 3.2. The necessary and sufficient conditions for the function given by Eq (2.8) to be part of the classes $M(\tau, \rho, \nu)$ and $N(\tau, \rho, \nu)$ are provided in Theorems 3.3 and 3.4. The impact of the Bessel function on the class $TN(\tau, \rho, \nu)$ is investigated in Theorems 3.5 and 3.6. The coefficient restriction proved in Theorem 3.2 is applied to investigate the impact of the hypergeometric function on the class $TN(\tau, \rho, \nu)$ in Theorems 3.7 and 3.8.

Theorem 3.1. Let $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $\nu \ge 0$ . A function f defined in (2.1) belongs to $TM(\tau, \rho, \nu)$ if and only if $$\sum_{n=2}^{\infty} [n(1+\nu) - (\rho+\nu)(1+n(2\tau^2-\tau) + (2\tau^2-3\tau))]|a_n| \le 1-\rho.$$

Corollary 3.1. For the parametric value $\tau = 0$ in Theorem 3.1, we derive Theorem 2.1 in [36], which asserts if $$\sum_{n=2}^{\infty} [n(1+\nu) - (\rho+\nu)]|a_n| \le 1-\rho$$ , then $f \in SD(\rho, \nu)$ .

Theorem 3.2. Let $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ . A function f given in (2.1) belongs to $TN(\tau, \rho, v)$ if and only if $$\sum_{n=2}^{\infty} n[n(1+\nu) - (\rho+\nu)(1+n(2\tau^2-\tau) + (2\tau^2-3\tau))]|a_n| \le 1-\rho.$$

Corollary 3.2. For the parametric value $\tau = 0$ in Theorem 3.2, we derive Theorem 2.2 in [36], which asserts if $$\sum_{n=2}^{\infty} n[n(1+\nu) - (\rho+\nu)]|a_n| \le 1 - \rho,$$ then $f \in Sk(\rho, \nu)$ . In the ensuing theorems, Lemma 2.1 is employed, and both the necessary and sufficient conditions for the function $z(2 - v_p(z))$ given by (2.8) to be categorized within the classes $TN(\tau, \rho, \nu)$ and $TM(\tau, \rho, \nu)$ , respectively, are given.

Theorem 3.3. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $\nu \ge 0$ , given the conditions where b < 0 and q > 0, it follows that $z(2 - \upsilon_p(z))$ , as described in (2.8), belongs to $TN(\tau, \rho, \nu)$ if and only if $$[1 - (2\tau^{2} - \tau)(\rho + \nu) + \nu]\upsilon_{p}''(1) + [2\nu - \rho - 2\tau(\rho + \nu)(4\tau^{2} - 3) + 3]\upsilon_{p}'(1)$$ $$+ [4\tau(\rho + \nu)(1 - \tau) + 1 - \rho][\upsilon_{p}(1) - 1] \le (1 - \rho)$$ (3.1) holds true.

Theorem 3.4. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ , given the conditions where b < 0 and m > 0, it follows that $z(2 - v_p(z))$ , given by (2.8), belongs to $TM(\tau, \rho, v)$ if and only if $$[1 - (2\tau^2 - \tau)(\rho + \nu) + \nu]v_p'(1) + [1 - (4\tau^2 - 4\tau)(\rho + \nu) - \rho][v_p(1) - 1] \le (1 - \rho)$$ (3.2) holds true.

Theorem 3.5. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ , consider b < 0, q > 0, and $f \in \Re^{\tau}(A, C)$ . If $$(A - C)|\tau|\{[(1 + \nu) - (\rho + \nu)(2\tau^2 - \tau)]\nu_p'(1) + [(1 - \rho) - (\rho + \nu)(4\tau^2 - 4\tau)][\nu_p(1) - 1]\} \le (1 - \rho),$$ (3.3) then $T(b,q)f(z) \in TN(\tau,\rho,\nu)$ .

Theorem 3.6. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ , given the conditions where b < 0 and m > 0, it follows that $\int_0^z (2 - v_p(t)) dt$ belongs to $TN(\tau, \rho, v)$ if and only if $$[1 + \nu - (\rho + \nu)(2\tau^2 - \tau)]v_p'(1) + [1 - \rho - (\rho + \nu)(4\tau^2 - 4\tau)][v_p(1) - 1] \le (1 - \rho). \tag{3.4}$$

Theorem 3.7. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ , if i, b > -1, ib < 0, and d > i + b + 2, then zF(z) belongs to $TN(\tau, \rho, v)$ if and only if $$\left[ (i)_2[(1+\nu) - (2\tau^2 - \tau)(\rho + \nu)](b)_2 + (d - i - b - 2)_2[1 - \rho - (4\tau^2 - 4\tau)(\rho + \nu)] + ib(d - i - b - 2)[-\rho - 2(4\tau^2 - 3)\tau(\rho + \nu) + 3 + 2\nu] \right] \ge 0.$$

Theorem 3.8. For $0 \le \rho < 1$ , $0 \le \tau < 1$ , and $v \ge 0$ , if d > i + b + 1 and i, b > 0, then (2z - zF(z)) belongs to the class $TN(\tau, \rho, v)$ if and only if $$\frac{\Gamma(d-i-b)\Gamma(d)}{\Gamma(d-b)\Gamma(d-a)} \left[ \frac{[3-2\tau(\rho+\nu)(4\tau^2-3)+2\nu-\rho]ib}{(1-\rho)(d-i-b-1)} + \frac{(i)_2[(1+\nu)-(2\tau^2-\tau)(\rho+\nu)](b)_2}{(1-\rho)(d-i-b-2)_2} + 1 \right] \leq 2.$$

Definition 2.1. For $0 \le \tau < 1$ and $v \ge 0$ , let $M(\tau, \rho, v)$ be the subset of $\mathbb{A}$ consisting of functions as given in (2.1) that meet the analytic criterion $$\Re\left\{\frac{zf'(z)}{4(\tau-\tau^{2})z+(2\tau^{2}-\tau)zf'(z)+(2\tau^{2}-3\tau+1)f(z)}-\rho\right\} > \nu\left|\frac{zf'(z)}{4(\tau-\tau^{2})z+(2\tau^{2}-\tau)zf'(z)+(2\tau^{2}-3\tau+1)f(z)}-1\right|, \quad z \in \mathbb{U}.$$ (2.9) If we set <span id="page-5-2"></span> $$\tau = \nu = 0$$ . it follows that $$\Re\left\{\frac{zf'(z)}{f(z)}\right\} > \rho,$$ which results in $$M(\tau, \rho, \nu) = S^*(\rho)$$ as defined in (2.3) (see [26]). If we assume $\tau = 0$ , it follows that $$\Re\left\{\frac{zf'(z)}{f(z)} - \rho\right\} > \nu \left|\frac{zf'(z)}{f(z)} - 1\right|,$$ leads to $$M(\tau, \rho, \nu) = SD(\rho, \nu)$$ as specified in ([36]). Observe that if $f \in SD(\rho, \nu)$ , then $f \in S^*(\frac{\rho-\nu}{1-\nu})$ , as illustrated in ([37]). Also, the previously mentioned class represents the generalized representation of various classes of starlike functions, which will be discussed immediately following Definition 2.2.

Definition 2.2. Let $N(\tau, \rho, \nu)$ be the subset of $\mathbb{A}$ consisting of functions as given in (2.1) that meet the analytic criterion $$\Re\left\{\frac{f'(z) + zf''(z)}{4(\tau - \tau^2) + (2\tau^2 - \tau)zf''(z) + (4\tau^2 - 4\tau + 1)f'(z)} - \rho\right\}$$ $$> \nu \left| \frac{f'(z) + zf''(z)}{4(\tau - \tau^2) + (2\tau^2 - \tau)zf''(z) + (4\tau^2 - 4\tau + 1)f'(z)} - 1 \right|.$$ (2.10) If one assumes <span id="page-5-3"></span> $$\tau = \nu = 0$$ it follows that $$\Re\left\{1+\frac{zf''(z)}{f'(z)}\right\} > \rho,$$ resulting in the class $M(\tau, \rho, v)$ being equivalent to $C(\rho)$ as defined in (2.4) ([26]). If we examine the case where $\tau = 0$ , then $$\Re\left\{\frac{zf'(z)}{f(z)} - \rho\right\} > \nu \left|\frac{zf'(z)}{f(z)} - 1\right|,$$ leading to $$M(\tau, \rho, \nu) = KD(\rho, \nu)$$ as defined in ([36]). It is important to recognize that if $f \in KD(\rho, \nu)$ , then it follows that $f \in K(\frac{\rho-\nu}{1-\nu})$ as illustrated in ([37]). Similarly, the previously mentioned class serves as the generalized representation of diverse subclasses of convex functions, which are elaborated upon below. From (2.9) and (2.10), it is obvious that $f \in N(\tau, \rho, \nu)$ if and only if $zf' \in M(\tau, \rho, \nu)$ . Moreover, let $$TM(\tau, \rho, \nu) = T \cap M(\tau, \rho, \nu)$$ and $$TN(\tau, \rho, \nu) = T \cap N(\tau, \rho, \nu).$$ Further, a straightforward computation shows that numerous subclasses of analytic functions introduced in different papers are particular cases of the subclasses $TM(\tau, \rho, \nu)$ and $TN(\tau, \rho, \nu)$ : In the case of $\tau=0$ , we identify the classes $TS_P(\rho,\nu)$ and $TS_P(\rho,\nu)$ which were considered by Bharati et al. in [38]. For $\tau=0$ and $\rho=0$ , we find the classes $TS_P(\nu)$ and $TK_P(\nu)$ , which were previously examined by Subramanian et al. in [39,40]. When $\tau=0$ and $\nu=1$ , we observe the classes $TS_P(\rho)$ and $TK_P(\rho)$ , studied by Bharati et al. in [38]. In the case of $\nu=0$ , we notice that $T^(\tau,\rho)$ and $C(\tau,\rho)$ are the classes previously investigated by Altintas et al. in [41]. For $\tau=0$ and $\nu=0$ , we uncover the classes $T^(\rho)$ and $C(\rho)$ , which were investigated earlier by Silverman in [26].