Theory of certain Non-Univalent Analytic functions

Theorem 1.1. [\[20\]](#page-17-4)(Covering theorem) Let min^|z|=^r <ψ(z) = ψ(−r). If f ∈ F(ψ) and f⁰ as defined in [\(1.5\)](#page-1-1), then either f is a rotation of f⁰ or $$\{w \in \mathbb{C} : |w| \le -f_0(-1)\} \subset f(\Delta),$$ where $$-f_0(-1) = \lim_{r \to 1} (-f_0(-r)).$$ Remark 1. The class F(ψ) gives a new insight and a pool of opportunities to study non-univalent functions. Several problems, namely Area estimates, Logarithmic coefficient bounds, sharp coefficient bounds, hankel determinant, Zalcman functional, and several radius problems like majorization, radius of convexity, Bohr-phenomenon, differential subordination implication, preservation under convolution and some famous integral etc. are yet to be known. Moreover, inclusion relation like $S^(\Psi) \subseteq S$ is not available for the class $\mathcal{F}(\psi)$ . This fact makes the above discussed problems non-trivial in $\mathcal{F}(\psi)$ . In fact, several implications which holds for $S^(\Psi)$ are not true for the class $\mathcal{F}(\psi)$ . For instance, if $f \in S^(\Psi)$ then $\Re(f(z)/z) > 0$ for all $z \in \Delta$ , see [28]. But $\Re(f(z)/z) > 0$ may not be true for $f \in \mathcal{F}(\psi)$ . See Figure 1. The function $$\hat{f}(z) = z \left( \frac{1 + z\sqrt{\beta}}{1 - z\sqrt{\beta}} \right)^{\frac{1}{2\sqrt{\beta}}} \in \mathcal{BS}(\beta) \Rightarrow \Re\left( \frac{\hat{f}(z)}{z} \right) > 0 \quad \text{for all } z \in \Delta.$$ However, for all $z \in \Delta$ , the function $$\kappa(z) := z \exp\left(\frac{z}{(1+\eta z)^2}\right) \in \mathcal{S}_1(\eta) \Rightarrow \Re\left(\frac{\kappa(z)}{z}\right) \geqslant 0, \quad \eta > 2 - \sqrt{3}.$$  FIGURE 1. Left side graph is for $\hat{f}(z)/z$ ( $\beta = 0.9$ ) and Right side graph is for $\kappa(z)/z$ ( $\eta = 0.9$ ). Likewise, Non-analytic functions with significant structure and properties similar to those satisfied by analytic functions exist as well. Non-analytic functions with such a nice structure are referred to as poly-analytic functions, which we shall discuss briefly soon in Section 2. For there interesting historical viewpoint and the application's side, we refer to read [1, Page 170] The version of Landau's theorem for poly-analytic functions is given in

Theorem 1.2. [1, Theorem 1] (Landau's theorem) Let $F(z) = \sum_{k=0}^{\alpha-1} \overline{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , $\alpha \geq 2$ on $\Delta$ , where $f_k$ are analytic such that $f_k(0) = 0$ , $f'_k(0) = 1$ and $|f_k(z)| \leq M$ , for all k, with M > 1. Then there is a constant $0 < \rho_1 < 1$ so that F is univalent in $|z| < \rho_1$ . In particular, $\rho_1$ satisfies $$1 - M\left(\frac{\rho_1(2 - \rho_1)}{(1 - \rho_1)^2} + \sum_{k=1}^{\alpha - 1} \frac{\rho_1^k(1 + k + k\rho_1)}{(1 - k\rho_1)^2}\right) = 0$$ and $F(\Delta_{\rho_1})$ contains a disk $\Delta_{R_1}$ , where $$R_1 = \rho_1 - {\rho_1}^2 \left( \frac{1 - {\rho_1}^{\alpha - 1}}{1 - \rho_1} \right) - M \sum_{k=0}^{\alpha - 1} \frac{{\rho_1}^{k+2}}{1 - \rho_1}.$$ For other relevant work in this direction, we refer to see [1, 9, 26] and references therein. For poly-analytic functions of order $\alpha$ given by $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ , where $f_k(z) = \sum_{n=0}^{\infty} a_{n,k} z^n$ , Abdulhadi and El Hajj [1] defined the majorant series M(F,r) by $$M(F,r) = \sum_{n=0}^{\infty} \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n},$$ and proved the following Bohr's type inequality

Theorem 1.3. [1, Theorem 2] Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic mappings for $k = 0, 1 \cdots, \alpha - 1$ , such that $g_k = f_0 + \bar{f}_k$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \cdots, \alpha - 1$ . Suppose that $f_0$ is univalent and normalized by $f_0(0) = 0$ , $f'_0(0) = 1$ . Then $$M(F,r) < 1$$ if $|z| < r_0$ , where $r_0 \approx 0.318$ is the minimum positive root of the polynomial $$(1-r)^3 - r + r^{\alpha+1} = 0.$$ The classical Bohr's inequality [6] says that if $f(z) = \sum_{n=0}^{\infty} a_n z^n$ is analytic in $\Delta$ and |f(z)| < 1 for all $z \in \Delta$ , then $\sum_{n=0}^{\infty} |a_n z^n| < 1$ for all z in $\Delta$ with |z| < 1/3. The constant 1/3 can not be improved and is known as Bohr's radius. In analogy with Bohr's inequality, there is also the notion of Rogosinski radius, however a little is known about Rogosinski radius as compared to Bohr radius, which states that, see [22, 33, 35]: if $g(z) = \sum_{k=0}^{\infty} b_k$ with |f(z)| < 1, then for every $N \ge 1$ we have $|\sum_{k=0}^{N-1} b_k z^k| \le 1$ for all $|z| \le 1/2$ . The radius 1/2 is called the Rogosinski radius. Kayumov et al. [17] studied a new quantity, called Bohr-Rogosinski sum and is given below: $$|g(z)| + \sum_{k=N}^{\infty} |b_k||z|^k, \quad |z| = r.$$ For the case N=1, note that this sum is similar to the Bohr's sum, where g(0) is replaced by |g(z)|. We also refer the readers to see [2, 5]. Now we say the family S(f) has Bohr-Rogosinski phenomenon, if there exists $r_N^f \in (0,1]$ such that the inequality: $$|g(z)| + \sum_{k=N}^{\infty} |b_k||z|^k \le |f(0)| + d(f(0), \partial\Omega)$$ holds for $|z| = r \leq r_N^f$ , where $d(f(0), \partial\Omega)$ denotes the Euclidean distance between f(0) and the boundary of $\Omega = f(\Delta)$ . The largest such $r_N^f$ is called the Bohr-Rogosinski radius. In this investigation, we obtain sharp initial coefficient's estimates along with some well known coefficient functionals for the class $\mathcal{F}(\psi)$ . We also derive Bohr and Bohr-Rogosinski radius for $\mathcal{F}(\psi)$ . Further, we explore the classes $\mathcal{F}(\psi)$ and $\mathcal{S}^*(\Psi)$ in connection with the poly-analytic functions to obtain Bohr and Rogosinski's type radius.

Lemma 2.1. ([34]). Let the analytic functions F and G be convex univalent in $\Delta$ . If $f \prec F$ and $g \prec G$ , then $$f q \prec F G \quad (z \in \Delta).$$ We also refer to see [28] for a parallel result concerning the class $\mathcal{S}^*(\Psi)$ .

Lemma 2.2. Let $\psi(\Delta)$ be a convex domain . If $f \in \mathcal{F}(\psi)$ . Then $f(z)/z \prec f_{\psi}(z)/z$ , where $f_{\psi}(z) = z \exp \int_0^z (\psi(t)/t) dt$ .

Lemma 2.3. [20] (Growth Theorem) Let us suppose that $\max_{|z|=r} \Re \psi(z) = \psi(r)$ and $\min_{|z|=r} \Re \psi(z) = \psi(-r)$ . Then $f \in \mathcal{F}(\psi)$ satisfies the sharp inequalities: $$r \exp\left(\int_0^r \frac{\psi(-t)}{t} dt\right) \le |f(z)| \le r \exp\left(\int_0^r \frac{\psi(t)}{t} dt\right), \quad (|z| = r). \tag{2.4}$$ Without loss of generality, we can have in Lemma 2.3 that $\max_{|z|=r} \Re \psi(z) = M(r)$ and $\min_{|z|=r} \Re \psi(z) = m(r)$ , where M and N are functions of r for each z in $\Delta$ such that |z|=r. Remark 3. In case of carathéodory functions p and q, we see that $p \prec q$ , where q(z) = (1+z)/(1-z) is the best dominant. But in case of functions in $\mathcal{S}$ , we have no such best dominant. Therefore, it is worthy to obtain upper bound for |f(z)| in general for the class $\mathcal{F}(\psi)$ , where $\psi \in \mathcal{S}$ . Note that $$\max_{|z|=r} \left| \frac{zf'(z)}{f(z)} \right| \le \frac{1-r+r^2}{(1-r)^2}.$$ This yields $$|f'(z)| \le \frac{(1-r+r^2)}{(1-r)^2} \exp\left(\frac{r}{1-r}\right)$$ and $|f(z)| \le r \exp\left(\frac{r}{1-r}\right)$ . The equality cases hold for the non-univalent function $$f(z) = z \exp\left(\frac{z}{1-z}\right).$$ At this conjunction in view of the remark 3, we propose the following: Question 1. Let $m(r) := \min_{|z|=r} |f(z)|$ , where $f \in \mathcal{F}(\psi)$ and $\psi \in \mathcal{S}$ . Find the value of m(r). In the following result, it is important to note that bounds for $|b_n|$ are not available. In rest of this section, we assume that coefficient $A_i$ of $\psi$ are positive. Remark 4. In case, when all the coefficients $A_i$ of function $\psi$ are not positive, then without loss of generality we can appropriately replace the function $f_{\psi}(z) = z \exp \int_0^z (\psi(t)/t) dt := z + \sum_{n=2}^{\infty} a_n z^n$ by the analytic function $\hat{f}_{\psi}(z) := z + \sum_{n=2}^{\infty} |a_n| z^n$ , throughout this paper.

Theorem 2.1. Let $\psi(\Delta)$ be convex. The class S(f), where $f \in \mathcal{F}(\psi)$ satisfies the Bohr-phenomenon $$\sum_{n=1}^{\infty} |b_n||z|^n \le d(0, \partial f(\Delta))$$ in $|z| = r \le \min\{1/3, r_0\}$ , where $r_0$ is least positive root of the equation $$f_{\psi}(r) - \exp \int_{0}^{1} \frac{\psi(-t)}{t} dt = 0.$$ (2.5) The result is sharp in case of $r_0 \leq 1/3$ .

Lemma 2.4. [14] Let $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and $g(z) = \sum_{n=0}^{\infty} b_n z^n$ be analytic in $\Delta$ . Let $\{\nu_k(r)\}_{k=0}^{\infty}$ be a sequence of non-negative functions, continuous in [0,1) such that the series $$\sum_{n=0}^{\infty} |b_n| \nu_k(r)$$ converges locally uniformly with respect to $r \in [0,1)$ . If $g \prec f$ , then $$\sum_{n=0}^{\infty} |a_n| \nu_n(r) \le \sum_{n=0}^{\infty} |b_n| \nu_n(r)$$ for all $|z| = r \leq \frac{1}{3}$ .

Theorem 2.2. (Generalized Bohr-Rogosinski sum). Let $\psi(\Delta)$ be convex. Let $\{\nu_n(r)\}_{n=1}^{\infty}$ be a non-negative sequence of continuous functions in [0,1] such that $$\nu_1(r) + \sum_{n=2}^{\infty} \left| \frac{f_{\psi}^{(n)}(0)}{n!} \right| \nu_n(r)$$ converges locally uniformly with respect to each $r \in [0,1)$ . If $$|f(z^m)| + \nu_1(r) + \sum_{n=2}^{\infty} \left| \frac{f_{\psi}^{(n)}(0)}{n!} \right| \nu_n(r) < \exp \int_0^1 \frac{\psi(-t)}{t} dt$$ and $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{F}(\psi)$ . Then $$|f(z^m)| + \sum_{n=1}^{\infty} |a_n| \nu_n(r) \le d(0, \partial\Omega)$$ (2.7) holds for $|z| = r \le r_b = \min\{1/3, r_0\}$ , where $m \in \mathbb{N}$ , $\Omega = f(\Delta)$ and $r_0$ is the smallest positive root of the equation: $$f_{\psi}(r^m) + \sum_{n=2}^{\infty} \left| \frac{f_{\psi}^{(n)}(0)}{n!} \right| \nu_n(r) = \exp \int_0^1 \frac{\psi(-t)}{t} dt - \nu_1(r),$$ where $$f_{\psi}(z) = z \exp \int_{0}^{z} \frac{\psi(t)}{t} dt.$$ Moreover, the inequality (2.7) also holds for the class S(f) in $|z| = r \le r_b$ . When $r_b = r_0$ , then the radius is best possible.

Corollary 2.2.1. Let $\psi(\Delta)$ be convex. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{F}(\psi)$ . Then $$|f(z^m)| + \sum_{n=N}^{\infty} |a_n||z|^n \le d(0, \partial\Omega)$$ (2.9) holds for $|z| = r \le r_b = \min\{1/3, r_0\}$ , where $m \in \mathbb{N}$ , $\Omega = f(\Delta)$ and $r_0$ is the smallest positive root of the equation: $$f_{\psi}(r^m) + \sum_{n=N}^{\infty} \left| \frac{f_{\psi}^{(n)}(0)}{n!} \right| r^n = \exp \int_0^1 \frac{\psi(-t)}{t} dt,$$ where $$f_{\psi}(z) = z \exp \int_0^z \frac{\psi(t)}{t} dt.$$ Moreover, the inequality (2.9) also holds for the class S(f) in $|z| = r \le r_b$ . When $r_b = r_0$ , then the radius is best possible. Here below, Corollary 2.2.1 yields the Bohr-Rogosinki radius for the class $\mathcal{BS}(\beta)$ .

Corollary 2.2.2. Let $0 < \beta \le 3 - 2\sqrt{2}$ . If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{BS}(\beta)$ . Then $$|f(z^m)| + \sum_{n=N}^{\infty} |a_n||z|^n \le d(0, \partial\Omega)$$ (2.10) holds for $|z| = r \le r_b = \min\{r_N^m(\beta), 1/3\}$ , where $m \in \mathbb{N}$ , $\Omega = f(\Delta)$ and $r_N^m(\beta)$ is the smallest positive root of the equation: $$\hat{f}_{\psi}(r^m) - S_{N-1}(\hat{f}) + \hat{f}(r) = \left(\frac{1 - \sqrt{\beta}}{1 + \sqrt{\beta}}\right)^{\frac{1}{2\sqrt{\beta}}},$$ where $$\hat{f}(z) = z \left( \frac{1 + z\sqrt{\beta}}{1 - z\sqrt{\beta}} \right)^{\frac{1}{2\sqrt{\beta}}}$$ and $S_{N-1}(\hat{f})$ denotes $(N-1)^{th}$ partial sum of $\hat{f}$ and $S_0 = 0$ . Moreover, the inequality (2.10) also holds for the class S(f) in $|z| = r \le r_b$ . The radius $r_N^m(\beta)$ is sharp with m = 1 for all $N \in \mathbb{N}$ . See, Figure 2 Remark 5. Taking $m \to \infty$ and $N \to 1$ in Corollary 2.2.2 gives [20, Theorem 4].  FIGURE 2. Existence of roots for different values of $\beta$ , m and N in Corollary 2.2.2. We now study poly-analytic functions in connection with the class $\mathcal{F}(\psi)$ . A continuous complexvalued function F defined in a domain $D \subseteq \mathbb{C}$ is poly-analytic of order $\alpha$ if it obeys the generalized Cauchy-Riemann equations $\partial^{\alpha} F/\partial \bar{z}^{\alpha} = 0$ . An iteration argument shows that F has the form $$F(z) = \sum_{k=0}^{\alpha - 1} \bar{z}^k f_k(z),$$ where $f_k$ are analytic functions for $k=0,\cdots,\alpha-1$ . Throughout, it is assumed that $\alpha\geq 2$ . In particular, a continuous complex-valued function F=u+iv in a domain $D\subseteq\mathbb{C}$ is said to be bi-analytic if $$\frac{\partial}{\partial \bar{z}} \left( \frac{\partial F}{\partial \bar{z}} \right) = 0.$$ Then F is a poly-analytic function of order 2. Note that $F_{\bar{z}}$ is analytic in D. In any simply connected domain D, it can be shown that F has the form $$F(z) = \bar{z}f_1(z) + f_0(z),$$ where $f_1$ and $f_0$ are analytic functions (and obviously $f_1 = F_{\bar{z}}$ ). Poly-analytic functions can have properties that differ from analytic functions, such as the ability to vanish on closed curves without vanishing identically. The function $F(z) = 1 - z\bar{z}$ is one such example. We obtain Theorem 1.3 for poly-analytic function with analytic counterparts from the classes $\mathcal{F}(\psi)$ , $\mathcal{S}^*(\Psi)$ and $\mathcal{C}(\Psi)$ for the Rogosinski's sum.

Theorem 2.3. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic such that $g_k = f_0 + \overline{f_k}$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \dots, \alpha - 1$ . Suppose that - (i) $L(r) = B_N^m(f_{\psi}, r)$ if $f_0 \in \mathcal{F}(\psi)$ and $\psi(\Delta)$ is convex - (ii) $L(r) = B_N^m(f_{\Psi}, r)$ if $f_0 \in \mathcal{S}^*(\Psi)$ (iii) $L(r) = B_N^m(g_{\Psi}, r)$ , where $zg'_{\Psi}(z) = f_{\Psi}(z)$ if $f_0 \in \mathcal{C}(\Psi)$ . Then $$B_N^m(F,r) = \sum_{n=N}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < \min\{1/3, r_0\}$$ where $N \in \mathbb{N}$ and $r_0$ is the minimal positive root of the equation $$L(r)(1 - r^{\alpha}) + r - 1 = 0$$

Corollary 2.3.1. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic such that $g_k = f_0 + \overline{f_k}$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \dots, \alpha - 1$ . Let $\beta \in (0, 3 - 2\sqrt{2}]$ and suppose that $f_0 \in \mathcal{BS}(\beta)$ . Then $$\sum_{n=1}^{m} \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < 1/3.$$

Corollary 2.3.2. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic such that $g_k = f_0 + \overline{f_k}$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \dots, \alpha - 1$ . Then $$B_N^m(F,r) = \sum_{n=N}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < \min\{1/3, r_0\}$$ where $N \in \mathbb{N}$ and $r_0$ is the minimal positive root of the equations: $$\left(\frac{r}{1-r}\right)(r^N-r^m)(1-r^\alpha)+r-1=0, \quad if \quad f_0 \in \mathcal{C}.$$ and $$\left(\frac{r}{1-r}\right) \left[\frac{r^N - r^m}{r(1-r)} - (m-1)r^m + (N-1)r^N\right] (1-r^\alpha) + r - 1 = 0, if \quad f_0 \in \mathcal{S}^*.$$ Remark 6. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ . Assume that $f_k \in \mathcal{C}$ for each $k = 0, \dots, \alpha - 1$ . Then the Rogosinski sum $$B_1^m(F,r) = \sum_{n=1}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1 \quad \text{in} \quad |z| < R_\alpha$$ where the Rogosinski radius $R_{\alpha}$ is the minimal positive root of the equations $$r(1 - r^{\alpha}) - (1 - r)^2 = 0.$$ Here, we note that the radius $R_{\alpha} > 1/3$ . Motivated from the Remark 6, we discuss the result when $f_k$ comes from the Janowski-starlike class.

Theorem 2.4. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ . Assume that $f_k \in \mathcal{S}^*[D, E]$ for each $k = 0, \dots, \alpha - 1$ . Then the Rogosinski sum $$B_1^m(F,r) = \sum_{n=1}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1 \quad in \quad |z| < R$$ for each $m \geq 2$ , where the Rogosinski radius R is the minimal positive root of the equations $$f_i(r)(1-r^{\alpha})+r-1=0,$$ where $$f_j(z) = \begin{cases} z(1+Ez)^{\frac{D-E}{E}}, & E \neq 0; \\ ze^{Dz}, & E = 0. \end{cases}$$ (2.12)

Corollary 2.4.1. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic such that $g_k = f_0 + \overline{f_k}$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \dots, \alpha - 1$ . Then $$B_1^m(F,r) = \sum_{n=1}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < \min\{1/3, r_0\}$$ where $r_0$ is the minimal positive root of the equations: $$\left(\frac{r}{1-r}\right)(r-r^m)(1-r^\alpha)+r-1=0, \quad if \quad f_0 \in \mathcal{C}.$$ and $$\left(\frac{r}{1-r}\right)\left[\frac{r-r^m}{r(1-r)} - (m-1)r^m\right](1-r^{\alpha}) + r - 1 = 0, \quad \text{if} \quad f_0 \in \mathcal{S}^*.$$ Remark 7. In Corollary 2.4.1, when $f_0 \in \mathcal{C}$ . Then for each value of $m, \alpha \geq 2$ , $$\sum_{n=1}^{m} \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1 \quad \text{in} \quad |z| < \frac{1}{3}.$$ When $f_0 \in \mathcal{S}^*$ , then $r_0 \leq 1/3$

Corollary 2.4.2. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k$ are analytic such that $g_k = f_0 + \overline{f_k}$ is sense-preserving in $\Delta$ and $f_k(0) = 0$ for each $k = 0, \dots, \alpha - 1$ . Then $$B_1^m(F,r) = \sum_{n=1}^{\infty} \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < \min\{1/3, r_0\}$$ where $r_0$ is the minimal positive root of the equations: $$(1-r)^2 - r^2 + r^{\alpha+2} = 0$$ , if $f_0 \in \mathcal{C}$ and $$(1-r)^3 - r + r^{\alpha+1} = 0$$ , if $f_0 \in \mathcal{S}^*$ . We now conclude this section with an interesting class studied by Bhowmik [7] on the basis of geometry, also see the family of Generalized Pascal Snails [15], which is as follows:

Theorem 2.5. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k \in Co(\beta)$ . Then $$B_N^m(F,r) = \sum_{n=N}^m \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < r_\beta(N)$$ where $N \in \mathbb{N}$ and $r_{\beta}(N)$ is the minimal positive root of the equation $$(1 - r^{\alpha})B_N^m(f_{\beta}, r) + r - 1 = 0,$$ where $f_{\beta}$ is given by (2.13).

Corollary 2.5.1. Let $F(z) = \sum_{k=0}^{\alpha-1} \bar{z}^k f_k(z)$ be a poly-analytic function of order $\alpha$ , where $f_k \in \mathcal{F}(f_1)$ . Then $$|F(z)| + \sum_{n=N}^{\infty} \sum_{k=0}^{\alpha-1} |a_{n,k}| r^{k+n} < 1, \quad \text{if} \quad |z| < \min\{r_0, 1/3\}$$ where $N \in \mathbb{N}$ and $r_0$ is the minimal positive root of the equation $$(1 - r^{\alpha})(r + r^{N}) - (1 - r)^{2} = 0.$$ As $\alpha \to \infty$ and N=2, then $r_0 \to 1/3$ and is best possible in this case. 3. Classical coefficient-functionals bounds for $\mathcal{F}(\psi)$ Suppose that the function $\psi$ is univalent in $\Delta$ with $\psi(0) = 0$ . Then we can write $$\psi(z) = A_1 z + A_2 z^2 + A_3 z^3 + \cdots$$ (3.1) Further, if the function $1 + \psi(z)$ belongs to the Carathéodory class then the Taylor's coefficients satisfies the well known bounds $|A_n| \leq 2$ . Otherwise, when $A_1 \neq 0$ then we have $|A_n|/|A_1| \leq n$ for each $n \in \mathbb{N}$ . Since the Taylor's coefficients of the function $f(z) = z + \sum_{k=2}^{\infty} a_k z^k$ in the class $\mathcal{F}(\psi)$ can be expressed in the form of $A_i$ , therefore to study the behavior of $|a_k|$ and some other coefficient functional, it is natural to consider the normalized form of f, that is $A_1 = 1$ . We start with some basic transformations which hold also for the class $\mathcal{S}$ of normalized univalent functions. Remark 9. Let $f \in \mathcal{F}(\psi)$ , where $\psi \in \mathcal{S}$ . Then the following transformations belong to $\mathcal{F}(\psi)$ : (i) Let $\alpha \in \mathbb{R}$ , $$e^{i\alpha}f(e^{i\alpha}z) = z + \sum_{n=2}^{\infty} a_n e^{i(n-1)\alpha}z^n.$$ (ii) Let $0 < t \le 1$ , $$\frac{1}{t}f(tz) = z + \sum_{n=2}^{\infty} a_n t^{n-1} z^n.$$ (iii) Let k is an positive integer $$[f(z^k)]^{\frac{1}{k}} = z + \frac{a_2}{k} z^{k+1} + \frac{1}{2k^2} (2ka_3 - (k-1)a_2^2) z^{2k+1} + \cdots$$ First, we obtain sharp initial coefficient's bounds, Zalcman [10, 27] and the Fekete-Szegö [3, 28] functional.

Theorem 3.1. If $f \in \mathcal{F}(\psi)$ , where $\psi \in \mathcal{S}$ . Then (1) sharp bounds for initial coefficient's are $$|a_2| \le A_1, \quad 2|a_3| \le \begin{cases} A_2 + A_1^2, & \text{if } A_1^2 + A_2 \ge A_1 \\ A_1, & \text{if } A_1 \ge A_1^2 + A_2 \ge -A_1 \\ -(A_2 + A_1^2), & \text{if } A_1^2 + A_2 \le -A_1 \end{cases}$$ and $$|a_4| \le \frac{A_1}{3} H(q_1, q_2), \quad where \quad \begin{cases} q_1 = \frac{4A_2 + 3A_1^2}{2A_1} \\ q_2 = \frac{2A_3 + A_1^3 + 3A_1A_2}{2A_1} \end{cases}$$ (2) sharp bound for the Zalcman functional of early coefficients is $$|a_2a_3 - a_4| \le \frac{A_1}{3}H(q_1, q_2), \quad \text{where} \quad \begin{cases} q_1 = \frac{A_1^2 + 4A_2}{2A_1} \\ q_2 = \frac{A_3 - A_1^2}{4A_1} \end{cases}$$ (3) sharp bound for the Fekete-Szegö functional is $$|a_3 - \nu a_2^2| \le \begin{cases} \frac{1}{2} (A_2 + (1 - 2\nu)A_1^2), & \text{if } A_1 + (2\nu - 1)A_1^2 \le A_2 \\ A_1/2, & \text{if } \frac{A_2 - A_1 + A_1^2}{2A_1^2} \le \nu \le \frac{A_2 + A_1 + A_1^2}{2A_1^2} \\ -\frac{1}{2} (A_2 + (1 - 2\nu)A_1^2), & \text{if } (2\nu - 1)A_1^2 \ge A_1 + A_2 \end{cases}$$ where $H(q_1, q_2)$ and $(q_1, q_2)$ are given by [32, Lemma 2].

Corollary 3.1.2. Let $f \in \mathcal{S}_{cs}(\beta)$ , $0 \le \beta < 1$ . Then (1) sharp bounds for initial coefficient's are $$|a_2| < 1$$ , $2|a_3| < 2 - \beta$ and $$|a_4| \le \frac{1}{3}H\left(\frac{7-4\beta}{2}, \frac{1+\beta^2}{1+\beta} + \frac{4-3\beta}{2}\right)$$ (2) sharp bound for the Zalcman functional of early coefficients is $$|a_2 a_3 - a_4| \le \frac{1}{3} H\left(\frac{5 - 4\beta}{2}, \beta(\beta - 1)\right)$$ (3) sharp bound for the Fekete-Szegö functional is $$2|a_3 - \nu a_2^2| \le \begin{cases} 2(1 - \nu) - \beta, & \text{if } 1 - 2\nu \ge \beta \\ 1, & \text{if } 1 - \beta \le 2\nu \le 3 - \beta \\ \beta - 2(1 - \nu), & \text{if } 2\nu \ge 3 - \beta \end{cases}$$ where $H(q_1, q_2)$ is given by [32, Lemma 2]. The equality cases follow for the extremal function $$f_{\beta}(z) = z \left(\frac{1+\beta z}{1-z}\right)^{\frac{1}{1+\beta}} = z + z^2 + \left(1 - \frac{\beta}{2}\right)z^3 + \frac{1}{6}(6 - 5\beta + 2\beta^2)z^4 + \cdots$$ Using the coefficients of the analytic function f given by $f(z) = z + \sum_{k=n+1}^{\infty} b_k z^k$ , Pommerenke, and Noonan and Thomas considered the Hankel determinant $H_q(n)$ which is defined by $$H_q(n) := \begin{vmatrix} b_n & b_{n+1} & \dots & b_{n+q-1} \\ b_{n+1} & b_{n+2} & \dots & b_{n+q} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n+q-1} & b_{n+q} & \dots & b_{n+2(q-1)} \end{vmatrix},$$ where $b_1 = 1$ . Estimation of upper bound of $|H_3(1)|$ [21, 23], $|H_2(2)|$ and $|H_2(1)|$ [3] for the functions belonging to various subclasses of univalent functions is a usual phenomenon in GFT, see [19]. For results on $\mathcal{S}^*(\Psi)$ , we refer to see [3, 4, 24, 27] We next tackle second hankel determinant $H_2(2) = (a_2a_4 - a_3^2)$ for the class $\mathcal{F}(\psi)$ : Here one should note that we have $|A_3| \leq 3$ , which is not the case in Ma-Minda function $\Psi$ .

Theorem 3.2. If $f \in \mathcal{F}(\psi)$ , where $\psi \in \mathcal{S}$ . (i) If $$|A_2| \le A_1$$ and $|A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2| \le 3A_1^2$ holds, then $|H_2(2)| < A_1^2/4$ . (ii) If either $$|A_2| \ge A_1$$ and $|A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2| \ge A_1|A_2| + 2A_1^2$ , or the conditions $$|A_2| \le A_1$$ and $|A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2| \ge 3A_1^2$ holds, then $$|H_2(2)| \leq \frac{|A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2|}{12}$$ (iii) If $|A_2| > A_1$ and $2A_1|A_2| + 2A_1^2 \ge |A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2|$ holds, then $$|H_2(2)| \le \frac{A_1^2}{4} - \frac{4A_1^2(|A_2| - A_1)^2}{48(|A_1^4 - 4A_1A_3 + 3A_2^2 + 6A_1^2A_2| - 2A_1|A_2| - A_1^2)}.$$

Corollary 3.2.1. Let $f \in \mathcal{BS}(\beta)$ , $0 \le \beta < 1$ . Then for all $\beta$ $$|H_2(2)| = |a_2a_4 - a_3^2| \le \frac{1}{4}.$$ The inequality is sharp with the extremal function $$f(z) = z \exp\left(\int_0^z \frac{t}{1-\beta t^4} dt\right) = z + \frac{z^3}{2} + \frac{z^5}{8} + \left(\frac{1+8\beta}{48}\right) z^7 + \cdots$$

Corollary 3.2.2. Let $f \in \mathcal{S}_{cs}(\beta)$ , $0 \le \beta < 1$ . Then $$|H_2(2)| = |a_2 a_4 - a_3^2| \le \frac{1}{4}, \quad \text{if} \quad |6 - 8\beta - \beta^2| \le 3$$ and $$|H_2(2)| \le \frac{|6 - 8\beta - \beta^2|}{12}, \quad \text{if} \quad |6 - 8\beta - \beta^2| \ge 3.$$ The equality in first case holds with the extremal function $$f(z) = z \exp\left(\int_0^z \frac{t}{1 + (\beta - 1)t^2 - \beta t^4} dt\right) = z + \frac{z^3}{2} + \left(\frac{3 - 2\beta}{8}\right) z^5 + \cdots$$ Further, application of this section can be seen for the non-Carathéodory regions given by Generalized Pascal Snails [15].

Definition 1. [\[20\]](#page-17-4) Let ψ be the analytic univalent function in ∆ such that ψ(0) = 0, ψ(∆) is starlike with respect to 0. Then $$\mathcal{F}(\psi) := \left\{ f \in \mathcal{A} : \left( \frac{zf'(z)}{f(z)} - 1 \right) \prec \psi(z), \ \psi(0) = 0 \right\}.$$ The following function plays the role of extremal $$f_0(z) = z \exp \int_0^z \frac{\psi(t)}{t} dt.$$ (1.5) Notice that for 1 + ψ(z) 6≺ (1 + z)/(1 − z), the functions in the class F(ψ) may not be univalent in ∆ which also implies F(ψ) 6⊆ S^∗ . Authors in [\[20\]](#page-17-4) noticed that such non-univalent analytic functions also exhibit nice geometric properties similar to class S ∗ (Ψ) such as

Definition 2. [7] A function $f: \Delta \to \mathbb{C}$ is said to be in the class of concave univalent functions with opening angle $\pi\beta$ , $\beta \in [1, 2]$ , at infinity if it fulfills the following conditions: - (a) f is analytic in $\Delta$ with the standard normalization and $f(1) = \infty$ . - (b) f maps $\Delta$ conformally onto a set whose complement with respect to $\mathbb{C}$ is convex. - (c) the opening angle of $f(\Delta)$ at $\infty$ is less than or equal to $\pi\beta$ . For the function $f \in Co(\beta)$ , the image domain $\mathbb{C}/f(\Delta)$ is closed and unbounded. Bhowmik [7] obtained sharp Coefficient's bounds, growth theorem, rotation theorem and distortion theorem etc. with the extremal function given by $$f_{\beta}(z) := \frac{1}{2\beta} \left[ \left( \frac{1+z}{1-z} \right)^{\beta} - 1 \right].$$ (2.13) It seems interesting to study as an independent interest the class defined by $$\mathcal{F}(f_{\beta}) := \left\{ f \in \mathcal{A} : \left( \frac{zf'(z)}{f(z)} - 1 \right) \prec f_{\beta}(z) \right\}.$$ For the class $Co(\beta)$ , Bhowmik and Das [8] studied the Bohr phenomenon. Now we explore this in connection with poly-analytic functions.