On the radius of concavity for certain classes of functions

Theorem 1. The radius of concavity $R_{Co(A)}$ of P' is (2.1) $$1 - \frac{2}{\sqrt{A+3}}, \quad A \in (1,2].$$ This radius is sharp.

Theorem 2. If $f \in P'(a)$ , then $\operatorname{Re} T_f(z) > 0$ for $|z| < R_{Co(A)}$ , where, $R_{Co(A)}$ is the least value of $r \in (0,1)$ satisfying $\phi(r) = 0$ with $$\phi(r) := r^4 - \frac{2(A+1)(1-a)}{A+3}r^3 + \frac{2(A(1-2a)-2a-3)}{A+3}r^2 - \frac{2(A(1-a)+3a+1)}{A+3}r + \frac{A-1}{A+3}.$$

Theorem 3. The radius of concavity $R_{Co(A)}$ of a linear-invariant family $\mathcal{F}$ of order $\alpha < \infty$ is (3.1) $$\frac{A+1+2\alpha-2\sqrt{(A+\alpha)(1+\alpha)}}{A-1}.$$ This radius is sharp.

Lemma 1. The function g(z) is analytic for |z| < 1 and satisfies g(0) = 1 and $\operatorname{Re} g(z) > 1/2$ for |z| < 1 if and only if $g(z) = 1/(1+z\phi(z))$ , where $\phi(z)$ is analytic and satisfies $|\phi(z)| \le 1$ for |z| < 1.

Theorem 4. If $f \in S^*(1/2)$ , then $\operatorname{Re} T_f(z) > 0$ for $|z| < R_{Co(A)}$ , where, $R_{Co(A)}$ is the least value of $r \in (0,1)$ satisfying u(r) = 0 with <span id="page-6-1"></span> $$u(r) = (A-1)r^3 - (3A+1)r^2 + (3A+7)r - A - 1.$$

Theorem 5. If $f \in \mathcal{U}_0(\lambda)$ , then Re $T_f(z) > 0$ for $|z| < R_{Co(A)}$ , where, $R_{Co(A)} = \min\{r_1, r_2\}$ with $$r_1 = \sqrt{\frac{5 + \lambda - \sqrt{(1 - \lambda)(25 - \lambda)}}{6\lambda}}$$ and $r_2$ being the least positive value of r satisfying $\phi(r) = 0$ with $$\phi(r) = -\lambda(9-A)r^3 - \lambda(A+11)r^2 - (A+3)r + A - 1.$$

Theorem 6. If f ∈ Vp(λ), then ReP^f (z) > 0 for |z| < RCo(p) , where, RCo(p) = min{r1, r2} with $$r_1 = \sqrt{\frac{3 - \lambda - \sqrt{(1 - \lambda)(9 - \lambda)}}{2\lambda}}$$ and r² being the least value of r ∈ (0, p) satisfying φ(r) = 0 with $$\phi(r) := (\lambda^2 p^2) r^5 + \lambda p (3\lambda p^2 - \lambda + 1) r^4 + \lambda (5\lambda p^2 - 4p^2 + 3) r^3 + p (1 - 4\lambda - \lambda p^2) r^2 - (1 + p^2 + 3\lambda p^2) r + p.$$ Proof. We know that from [\[4,](#page-12-11) Section 2] if ^f ∈ Vp(λ), then there exists a holomorphic function ^w¹ such that ^w1(D) ^⊆ ^D and <span id="page-9-0"></span>(4.1) $$\frac{z}{f(z)} = 1 - \left(\frac{f''(0)}{2}\right)z + \lambda z \int_0^z w_1(t)dt, \quad z \in \mathbb{D}.$$ Let <span id="page-9-2"></span>(4.2) $$w(z) := \left(\int_{p}^{z} w_{1}(t)dt\right)/(z-p), \quad z \in \mathbb{D}.$$ Then from [\[3,](#page-12-12) Theorem 1], we get (4.3) $$\frac{z}{f(z)} = \frac{-(z-p)(1-\lambda pzw(z))}{p}, \quad z \in \mathbb{D},$$ where ^w is analytic in ^D and ^|w(z)| ≤ 1, ^z ^∈ ^D. By differentiating [\(4.1\)](#page-9-0) with respect to z, we get <span id="page-9-1"></span> $$-z\left(\frac{z}{f(z)}\right)' + \frac{z}{f(z)} = 1 - \lambda z^2 w_1(z), \quad \text{for } z \in \mathbb{D}.$$ This implies that $$\left(\frac{z}{f(z)}\right)^2 f'(z) = 1 - \lambda z^2 w_1(z), \quad z \in \mathbb{D}.$$ Then from [\(4.3\)](#page-9-1), we get $$f'(z) = \frac{p^2(1 - \lambda z^2 w_1(z))}{(z - p)^2(1 - \lambda pzw(z))^2}, \quad z \in \mathbb{D}.$$ Thus, we have for $z \in \mathbb{D}$ , <span id="page-10-0"></span> $$(4.4) \frac{zf''(z)}{f'(z)} = -\frac{2z}{z-p} - \frac{\lambda z^2 (2w_1(z) + zw_1'(z))}{1 - \lambda z^2 w_1(z)} + \frac{2\lambda pz(w(z) + zw_1'(z))}{1 - \lambda pzw(z)}, \quad z \in \mathbb{D}.$$ From (4.2), we get $$(z-p)w'(z) + w(z) = w_1(z).$$ Then by a simple calculation we see that $$zw'(z) + w(z) = \frac{1}{z-p}(zw_1(z) - pw(z)).$$ By using this, we deduce from (4.4) that for $z \in \mathbb{D}$ , <span id="page-10-1"></span> $$(4.5) 1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p} = -\frac{\lambda z^2 (2w_1(z) + zw_1'(z))}{1 - \lambda z^2 w_1(z)} + \frac{2\lambda pz (zw_1(z) - pw(z))}{(z-p)(1 - \lambda pzw(z))}.$$ Let us define $$u(z) = z^2 w_1(z), \quad z \in \mathbb{D}.$$ Then we get u(0) = 0 = u'(0) and $|u(z)| \le |z|^2$ , $z \in \mathbb{D}$ . The equation (4.5) becomes <span id="page-10-2"></span>(4.6) $$1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p} = -\frac{\lambda z u'(z)}{1 - \lambda z u(z)} + \frac{2\lambda p(u(z) - pzw(z))}{(z-p)(1 - \lambda pzw(z))}.$$ Since $w_2(z) = (u(z))/z$ is an analytic function from $\mathbb{D}$ to $\mathbb{D}$ , we get $$|w_2'(z)| \le \frac{1 - |w_2(z)|^2}{1 - |z|^2}, \quad z \in \mathbb{D},$$ which is equivalent to $$|zu'(z) - u(z)| \le \frac{|z|^2 - |u(z)|^2}{1 - |z|^2}, \quad z \in \mathbb{D}.$$ From (4.6) and the above inequality, we get $$\left|1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p}\right| \le \frac{\lambda(|z|^2 - |u(z)|^2)}{(1-|z|^2)(1-\lambda|u(z)|)} + \frac{\lambda|u(z)|}{(1-\lambda|u(z)|)} + \frac{2\lambda p(|u(z)| + p|z|)}{||z| - p|(1-\lambda p|z|)},$$ for $z \in \mathbb{D}$ . If we let |u(z)| = x and |z| = r (note that $0 \le x \le r^2$ ), the above inequality becomes <span id="page-10-3"></span> $$(4.7)$$ Let us denote $$\phi(x) = \frac{\lambda(r^2 - x^2)}{(1 - r^2)(1 - \lambda x)} + \frac{\lambda x}{1 - \lambda x} + \frac{2\lambda p(x + pr)}{(p - r)(1 - \lambda pr)}, \quad \text{for} \quad 0 \le x \le r^2.$$ Then by some simple computations, we see that $$\phi(x) \le \phi(r^2) = \frac{2\lambda r^2}{1 - \lambda r^2} + \frac{2\lambda pr(p+r)}{(p-r)(1 - \lambda pr)}, \quad \text{if} \quad r < r_1,$$ where $r_1$ is given in the statement of the theorem. Thus from (4.7), we get <span id="page-11-1"></span>(4.8) $$\left| 1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p} \right| \le \frac{2\lambda r^2}{1-\lambda r^2} + \frac{2\lambda pr(p+r)}{(p-r)(1-\lambda pr)}, \quad |z| = r < r_1.$$ This implies that <span id="page-11-0"></span>(4.9) Re $$\left(1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p}\right) \le \frac{2\lambda r^2}{1-\lambda r^2} + \frac{2\lambda pr(p+r)}{(p-r)(1-\lambda pr)}, \quad |z| = r < r_1.$$ It is easy to see that Re $$\left(\frac{1+pz}{1-pz}\right) \ge \frac{1-pr}{1+pr}, \quad |z| = r < 1.$$ Applying (4.9) and the above inequality, we get $$\operatorname{Re} P_f(z) \ge \frac{1 - pr}{1 + pr} - \frac{2\lambda r^2}{1 - \lambda r^2} - \frac{2\lambda pr(p+r)}{(p-r)(1 - \lambda pr)}, \quad |z| = r < r_1.$$ The right hand side of the above inequality is strictly positive if $|z| = r < r_2$ , where $r_2$ is defined in the statement of the theorem. Thus, $\operatorname{Re} P_f(z) > 0$ if $|z| < R_{Co(p)}$ , where, $R_{Co(p)} = \min\{r_1, r_2\}$ . We now investigate the existence of $r_2$ for each $\lambda \in (0, 1]$ and $p \in (0, 1)$ . The function $\phi$ which is defined in the statement of the theorem, is continuous on [0, p] with $$\phi(0) = p > 0$$ and $\phi(p) = -4\lambda p^3(1+p^2)(1-\lambda p^2) < 0$ . Therefore, by the intermediate value theorem, $\phi$ has at least one root in (0, p). This establishes the existence of $r_2$ for every $\lambda \in (0, 1]$ and $p \in (0, 1)$ . In the following Corollary, we find radius of convexity for the class $\mathcal{V}_p(\lambda)$ by applying the estimate derived in (4.8).

Corollary 1. If $f \in \mathcal{V}_p(\lambda)$ , then f maps $|z| < R_c$ onto a convex set, where, $R_c = \min\{r_1, r_2\}$ with $$r_1 = \sqrt{\frac{3 - \lambda - \sqrt{(1 - \lambda)(9 - \lambda)}}{2\lambda}}$$ and $r_2$ being the least value of $r \in (0, p)$ satisfying $\psi(r) = 0$ with $$\psi(r) = (\lambda^2 p) r^5 + \lambda (1 + 2\lambda p^2) r^4 - \lambda p (1 - 5\lambda p^2) r^3 + (1 - 5\lambda p^2) r^2 - p (2 + 3\lambda p^2) r + p^2.$$

Definition 1.1. The radius of concavity (with respect to Co(A)) of a subset $A_1$ of A is the largest number $R_{Co(A)} \in (0,1]$ such that for each function $f \in A_1$ , $\operatorname{Re} T_f(z) > 0$ for all $|z| < R_{Co(A)}$ , where $T_f$ is defined in (1.1). For meromorphic functions, the definition of radius of concavity is not the same as above. Again we recall it from [2]. Let $\mathcal{A}(p)$ be the class of meromorphic functions in $\mathbb{D}$ with a simple pole at $z=p,\ p\in(0,1)$ and normalized by the condition f(0)=0=f'(0)-1. Let S(p) denote the set of all univalent functions in $\mathcal{A}(p)$ . Let Co(p) be the class of functions $f\in S(p)$ such that $\overline{\mathbb{C}}\setminus f(\mathbb{D})$ is a bounded convex set. In 1971, J. A. Pfaltzgraff and B. Pinchuk (see [13, p. 145]) proved that $f\in Co(p)$ if and only if $f\in S(p)$ such that <span id="page-1-1"></span>Re $$P_f(z) > 0$$ , $z \in \mathbb{D}$ , $P_f(p) = \frac{1+p^2}{1-p^2}$ and $P_f(0) = 1$ , where (1.2) $$P_f(z) := -\left[1 + \frac{zf''(z)}{f'(z)} + \frac{z+p}{z-p} - \frac{1+pz}{1-pz}\right].$$ For meromorphic functions, the radius of concavity (with respect to Co(p)) for a subclass of $\mathcal{A}(p)$ is defined as below (see [2, Definition 1.1]).

Definition 1.2. The radius of concavity (with respect to Co(p)) of a subset $\mathcal{A}_1(p)$ of $\mathcal{A}(p)$ is the largest number $R_{Co(p)} \in (0,1]$ such that for each function $f \in \mathcal{A}_1(p)$ , $\operatorname{Re} P_f(z) > 0$ for all $|z| < R_{Co(p)}$ , where, $P_f$ is defined in (1.2). We mention here that, in [2, Theorem 4] we obtained the radius of concavity of S. We also derived a lower bound for the radius of concavity of S(p) (see [2, Theorem 2]). Furthermore, we determined the same for the linear combinations of functions belonging to S(p), S, and Co(A) with complex coefficients. Let P' be the class of functions $f \in \mathcal{A}$ such that $\operatorname{Re} f'(z) > 0$ , $z \in \mathbb{D}$ . In 1962, Macgregor (c.f. [8]) determined the radius of convexity of the class P' as $\sqrt{2} - 1$ with an extremal function $f_0(z) = -z - 2\ln(1-z)$ . In 2004, Todorov (c.f. [18]) gave an alternative proof of this result and generalized all the extremal functions associated with it. Todorov also considered the class P' with fixed second coefficient and obtained the radius of convexity for the same. In this article, we will compute the radius of concavity of P' using Definition 1.1. Furthermore, we will determine a lower bound for the radius of concavity of the class P' with fixed second coefficient. Let $\mathcal{LS}$ be the set of locally univalent functions defined in $\mathbb{D}$ , with the normalization f(0) = 0 = f'(0) - 1. Let $\operatorname{Aut}(\mathbb{D})$ denote the set of holomorphic automorphisms of $\mathbb{D}$ . If $f \in \mathcal{LS}$ and $\phi \in \operatorname{Aut}(\mathbb{D})$ , then the Koebe transform of f with respect to $\phi$ is defined as $$\Lambda_{\phi}(f)(z) := \frac{(f \circ \phi)(z) - (f \circ \phi)(0)}{(f \circ \phi)'(0)}, \quad z \in \mathbb{D}.$$ We recall that a family $\mathcal{F}$ is called a linear-invariant family (L.I.F.) if $\mathcal{F} \subset \mathcal{LS}$ and $\Lambda_{\phi}(f) \in \mathcal{F}$ for all $f \in \mathcal{F}$ , $\phi \in \operatorname{Aut}(\mathbb{D})$ . The order of a L.I.F. $\mathcal{F}$ is ord $$\mathcal{F} := \sup \left\{ \left| \frac{f''(0)}{2} \right| : f \in \mathcal{F} \right\}.$$ For a detailed study of this linear-invariant family, we urge the reader to go through [5, Ch. 5]. In 1964, Pommerenke (c.f. [14]) proved that the radius of convexity of a linear-invariant family $\mathcal{F}$ is $\alpha - \sqrt{\alpha^2 - 1}$ , where ord $\mathcal{F} = \alpha < \infty$ . In this article, we will compute the radius of concavity of a linear-invariant family $\mathcal{F}$ . As every convex function is starlike, a function that is convex in $\mathbb{D}$ , satisfies Re (zf'(z)/f(z)) > 0 for $z \in \mathbb{D}$ . In fact, for $f \in C$ , the aforementioned condition can be improved to Re (zf'(z)/f(z)) > 1/2 for $z \in \mathbb{D}$ (see [17]). In other words, if $f \in C$ , then $f \in S^(1/2)$ . In [9], Macgregor obtained the radius of convexity of the class $S^(1/2)$ as $\sqrt{2\sqrt{3}-3}$ . In this article, we will determine a lower bound for the radius of concavity of $S^*(1/2)$ . In the final section of this article, we investigate radius of concavity for another interesting subclass $\mathcal{U}(\lambda)$ , $\lambda \in (0,1]$ of $\mathcal{S}$ . We first present a brief overview about this class of functions. For $\lambda \in (0,1]$ , let $$\mathcal{U}(\lambda) = \{ f \in \mathcal{A} : |U_f(z)| < \lambda, \ z \in \mathbb{D} \},$$ where $U_f(z) := (z/f(z))^2 f'(z) - 1$ . We know that $\mathcal{U}(\lambda) \subset \mathcal{S}$ , for $0 < \lambda \leq 1$ , (c.f. [12]). For more details about this class, we urge the reader to go through the articles [11, 12, 16] and references therein. Let $\mathcal{U}_0(\lambda)$ be the class of functions f in $\mathcal{U}(\lambda)$ with the additional condition f''(0) = 0. In [15], Ponnusamy and Vasundhra proved that $\mathcal{U}_0(\lambda) \subset S$ for $0 < \lambda \leq 1/\sqrt{2}$ and $\mathcal{U}_0(\lambda) \subset S^(1/2)$ for $0 < \lambda \leq 1/3$ . They also proposed a conjecture that each $f \in \mathcal{U}_0(\lambda)$ is convex in $\mathbb{D}$ if $0 < \lambda \leq 3 - 2\sqrt{2}$ . In [19], Vasundhra also derived a lower bound for the radius of convexity of $\mathcal{U}(\lambda)$ and $\mathcal{U}_0(\lambda)$ . In [7], it is shown that the above conjecture is not valid and an improved lower bound for the radius of convexity of $\mathcal{U}_0(\lambda)$ has been found. In this article, we will determine a lower bound for the radius of concavity of the class $\mathcal{U}_0(\lambda)$ . In [4], the first author of this article and F. Parveen considered a meromorphic analog of the class $\mathcal{U}(\lambda)$ as follows. For $\lambda \in (0,1]$ , let $$\mathcal{V}_p(\lambda) = \{ f \in \mathcal{A}(p) : |U_f(z)| < \lambda, \ z \in \mathbb{D} \}.$$ We urge the reader to go through the articles [3, 4] for more information about this class of functions. Finally, we will obtain a lower bound for the radius of concavity of the class $\mathcal{V}_p(\lambda)$ . As a byproduct of this result, we also find radius of convexity for functions in the class $\mathcal{V}_p(\lambda)$ . It is worth to mention here that, in certain cases, the obtained radii may not be sharp; and determining the exact radii will be interesting problems of study in future.