On the logarithmic coefficients of Ma-Minda type convex functions

Proposition 1. The function $\psi(z)$ defined by (2.2) is not convex when $c \in (1/2, 2)$ . On the other hand, by numerical computation, we demonstrate that $\psi(z)$ defined by (2.2) fails to be convex for several values of c in (0, 1/2]. To show that $\psi(z)$ is not convex for a certain value of $c \in (0, 1/2]$ , we show that $$\operatorname{Re}\Psi(e^{i\theta}) = \operatorname{Re}\left(1 + \frac{e^{i\theta}\psi''(e^{i\theta})}{\psi'(e^{i\theta})}\right) < 0$$ for some values of $z = e^{i\theta}$ , $0 \le \theta < 2\pi$ on the unit circle |z| = 1. In Table 1, we demonstrate that $\psi(z)$ is not convex for selected values of $c \in (0, 1/2]$ . Consequently, for the values of c mentioned in Table 1, the function $\psi(z)$ fails to be convex. This suggests that $\psi(z)$ may lack convexity for the entire interval $c \in (0, 1/2]$ , thereby implying that the estimate (2.3) is not generally valid. Here it is pertinent to mention that Ponnusamy et al. [17] established sharp estimates for $|\gamma_n|$ when n = 1, 2, 3 for all $c \in (0, 3]$ , while obtaining sharp bounds for $|\gamma_4|$ and $|\gamma_5|$ only under the restricted conditions $c \le 144/55$ and $c \le 80/61$ , respectively. | c | θ<br>(0<br>≤<br>θ <<br>2π) | iθ)<br>Re Ψ(e | |------|----------------------------|---------------------| | 0.1 | 10−20)π<br>(2<br>− | 106<br>−1.66<br>× | | 0.15 | 10−31)π<br>−<br>(2 | 1024<br>−1.02<br>× | | 0.2 | 10−28)π<br>(2<br>− | 1022<br>−1.077<br>× | | 0.25 | 10−13)π<br>(2<br>− | −375.774 | | 0.3 | 10−30)π<br>(2<br>− | 1031<br>−9.57<br>× | | 0.35 | 10−17)π<br>−<br>(2 | 1012<br>−4.65<br>× | | 0.4 | 10−20)π<br>(2<br>− | 1019<br>−2.05<br>× | | 0.45 | 10−19)π<br>(2<br>− | 1019<br>−3.41<br>× | | 0.4 | 10−20)π<br>−<br>(2 | 1019<br>−2.05<br>× | | 0.5 | 10−20)π<br>(2<br>− | 1013<br>−2.18<br>× | Table 1. Later on, Allu et al. [\[4\]](#page-17-17) also studied the logarithmic coefficients of functions in F(c) extending the results of Cho et al. [\[3\]](#page-17-15) and proved the following logarithmic coefficient inequalities. <span id="page-4-0"></span>Theorem B. [\[4\]](#page-17-17) Let f ∈ F(c) for c ∈ (0, 0.656] ∪ {2}. Then the logarithmic coefficients γⁿ of f satisfy the inequalities $$\sum_{n=1}^{\infty} n^2 |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} |D_n|^2,$$ and $$\sum_{n=1}^{\infty} (n+1)^t |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} |D_n|^2 \quad \text{for } t \le 2,$$ where Dⁿ are the Taylor's coefficients of ψ(z) given by [\(2.2\)](#page-3-0). The first inequality is sharp. In the proof of Theorem [B,](#page-4-0) the authors of [\[4\]](#page-17-17) used the convexity of the function ψ(z) defined by [\(2.2\)](#page-3-0) which compelled them to restrict the result for c ∈ (0, 0.656]∪ {2}. But after a close inspection, we observed that the convexity of ψ(z) is not necessary. In the next theorem, we extend Theorem [B](#page-4-0) with c ∈ (0, 2]. <span id="page-4-1"></span>Theorem 2.1. Let f ∈ F(c) for c ∈ (0, 2]. Then the logarithmic coefficients γⁿ of f satisfy the inequalities $$\sum_{n=1}^{\infty} n^2 |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} |D_n|^2,$$ and $$\sum_{n=1}^{\infty} (n+1)^t |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} |D_n|^2 \quad \text{for } t \le 2,$$ where the Dⁿ are the Taylor coefficients of ψ(z) given by [\(2.2\)](#page-3-0). Moreover, the first inequality is sharp. We note that if we take $\alpha = 1 - c/2$ , then the class $\mathcal{F}(c)$ for $c \in (0, 2]$ reduces to the standard class $\mathcal{C}(\alpha)$ of convex functions of order $\alpha$ , where $\alpha \in [0, 1)$ . 2.2. The class C(A, B). For $-1 \leq B < A \leq 1$ , the class C(A, B) of Janowski convex functions consists of functions $f \in A$ satisfying the subordination relation $$1 + \frac{zf''(z)}{f'(z)} \prec \frac{1 + Az}{1 + Bz}.$$ It is an easy exercise to see that the function $K_{A,B}$ defined by <span id="page-5-2"></span>(2.5) $$K_{A,B}(z) = \begin{cases} \frac{1}{A} \left( (1 + Bz)^{A/B} - 1 \right), & A \neq 0, B \neq 0, \\ \frac{1}{B} \log(1 + Bz), & A = 0, \\ \frac{1}{A} \left( e^{Az} - 1 \right), & B = 0, \end{cases}$$ is a function in $\mathcal{C}(A,B)$ . Using the function, we define another function $\psi_{A,B}$ as <span id="page-5-0"></span>(2.6) $$\psi_{A,B} = \frac{zK'_{A,B}(z)}{K_{A,B}(z)} =: 1 + \sum_{n=1}^{\infty} E_n z^n.$$ It is worth noting that Cho et al. [3, Theorem 2.2] studied the logarithmic coefficients for the class C(0, B) where $B \in [-0.99, 0)$ , establishing results analogous to Theorem A. <span id="page-5-1"></span>Theorem C. [3, Theorem 2.2] Let $f \in \mathcal{C}[0, B]$ for $-0.99 \leq B < 0$ . Then, the logarithmic coefficients $\gamma_n$ of f satisfy the inequalities $$|\gamma_n| \le \frac{|E_n|}{2n}, \quad n \in \mathbb{N},$$ and $$\sum_{n=1}^{\infty} |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{|E_n|^2}{n^2},$$ where $E_n$ are the Taylor coefficients of $\psi_{0,B}$ defined by (2.6) and the inequalities are sharp. The proof of Theorem C relies on the convexity of $\psi_{0,B}(z)$ defined in (2.6). The authors claimed to verify this property pictorially (see [3, Fig. 2]) using MAPLE^TM. From the first inequality of Theorem C, it follows that $$|\gamma_1| \le \frac{|B|}{4}, \quad |\gamma_2| \le \frac{5|B|^2}{48}, \quad |\gamma_3| \le \frac{|B|^3}{16}$$ for functions in the class C[0, B] with $-0.99 \le B < 0$ . However, these estimates are not universally correct, as we demonstrate in Theorem 2.2. This suggests that the underlying claim regarding the convexity of $\psi_{0,B}(z)$ is incorrect. In the next theorem, we obtain sharp estimates of initial logarithmic coefficients for functions in the class C(A, B).

Theorem 2.2. Let $f \in C(A, B)$ with $-1 \leq B < A \leq 1$ . Then the logarithmic coefficients $\gamma_n$ of f(z) satisfy $$|\gamma_{1}| \leq \frac{A - B}{4},$$ $$|\gamma_{2}| \leq \begin{cases} \frac{A - B}{12} & for |A - 5B| \leq 4, \\ \frac{A - B}{48} |A - 5B| & for |A - 5B| > 4, \end{cases}$$ $$|\gamma_{3}| \leq \begin{cases} \frac{A - B}{24}, & for (\mu, \nu) \in D_{1} \cup D_{2}, \\ \frac{A - B}{48} |B(A - 3B)|, & for (\mu, \nu) \in D_{6}, \\ \frac{A - B}{72\sqrt{3}} \cdot \frac{(|A - 5B| + 2)^{3/2}}{\sqrt{|A - 5B| + 2 + 3B^{2} - AB}} & for (\mu, \nu) \in D_{8} \cup D_{9}, \end{cases}$$ where $\mu = \frac{1}{2}(A-5B)$ and $\nu = \frac{1}{2}(3B^2-AB)$ and the sets $D_1, D_2, D_6, D_8, D_9$ are defined as in Lemma 3.1. All the estimates are sharp. On the other hand, Allu and Sharma [4] proved logarithmic coefficient inequalities similar to Theorem B for functions in the classes C(0, B) with $B \in [-0.99, 0)$ and C(A, 0) with $A \in (0, 1]$ . In the next theorem, we prove logarithmic coefficient inequalities similar to Theorem B for functions in the class C(A, B) with $-1 \le B < A \le 1$ , which generalize the previous results of Allu and Sharma [4].

Theorem 2.3. Let $f \in C(A, B)$ with $-1 \leq B < A \leq 1$ . Then the logarithmic coefficients $\gamma_n$ of f satisfy the inequalities $$\sum_{n=1}^{\infty} n^2 |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} |E_n|^2,$$ and $$\sum_{n=1}^{\infty} (n+1)^t |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} |E_n|^2 \quad \text{for } t \le 2,$$ where the coefficients $E_n$ are defined as in (2.6). Moreover, the first inequality is sharp. 2.3. The class $S_{\alpha}$ . For $-\pi/2 < \alpha < \pi/2$ , the class $S_{\alpha}$ , sometimes known as the Robertson class, is defined as follows $$S_{\alpha} =: \left\{ f \in \mathcal{A} : \operatorname{Re} \left\{ e^{i\alpha} \left( 1 + \frac{zf''(z)}{f'(z)} \right) \right\} > 0 \right\}.$$ For the class $S_{\alpha}$ , we obtain the following counterpart of Theorem 2.3.

Theorem 2.4. If $f \in \mathcal{S}_{\alpha}$ ( $|\alpha| < \pi/2$ ), then the logarithmic coefficients $\gamma_n$ of f satisfy the inequalities $$\sum_{n=1}^{\infty} n^2 |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} |F_n|^2,$$ and $$\sum_{n=1}^{\infty} (n+1)^t |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} |F_n|^2 \quad \text{for } t \le 2,$$ where the $F_n$ are Taylor coefficients of the function $$\psi_2(z) = \frac{Az}{(1-z)(1-(1-z)^A)} =: 1 + \sum_{n=1}^{\infty} F_n z^n,$$ with $A = e^{-2i\alpha}$ . Moreover, the first inequality is sharp. In the next theorem, we obtain sharp estimates of the initial logarithmic coefficients for functions in the class $S_{\alpha}$ .

Theorem 2.5. If $f \in \mathcal{S}_{\alpha}$ for $|\alpha| < \pi/2$ , then the logarithmic coefficients $\gamma_n$ of f(z)satisfy $$|\gamma_1| \le \frac{\cos \alpha}{2}, \quad |\gamma_2| \le \frac{\cos \alpha}{6} \sqrt{4 + 5\cos^2 \alpha}, \quad |\gamma_3| \le \frac{\cos \alpha}{3} \sqrt{1 + 3\cos^2 \alpha}.$$ All the estimates are sharp.

Lemma 3.1. [20] Let $f \in \mathcal{B}_0$ be of the form (3.1). Then for real $\mu$ and $\nu$ , we have $$|c_{3} + \mu c_{1}c_{2} + \nu c_{1}^{3}| \leq \begin{cases} 1 & \text{if } (\mu, \nu) \in D_{1} \cup D_{2} \cup \{(2, 1)\} \\ |\nu| & \text{if } (\mu, \nu) \in \bigcup_{k=3}^{7} D_{k} \end{cases}$$ $$\frac{2}{3} (|\mu| + 1) \left( \frac{|\mu| + 1}{3(|\mu| + 1 + \nu)} \right)^{1/2} & \text{if } (\mu, \nu) \in D_{8} \cup D_{9}$$ $$\frac{1}{3} \nu \left( \frac{\mu^{2} - 4}{\mu^{2} - 4\nu} \right) \left( \frac{\mu^{2} - 4}{3(\nu - 1)} \right)^{1/2} & \text{if } (\mu, \nu) \in D_{10} \cup D_{11} - \{(2, 1)\} \end{cases}$$ $$\frac{2}{3} (|\mu| - 1) \left( \frac{|\mu| - 1}{3(|\mu| - 1 - \nu)} \right)^{1/2} & \text{if } (\mu, \nu) \in D_{12}.$$ $$where$$ where $$D_1 = \left\{ (\mu, \nu) : |\mu| \le \frac{1}{2}, -1 < \nu \le 1 \right\},$$ $$\begin{split} D_2 &= \left\{ (\mu, \nu) : \frac{1}{2} \leq |\mu| \leq 2, \ \frac{4}{27} (|\mu| + 1)^3 - (|\mu| + 1) \leq \nu \leq 1 \right\}, \\ D_3 &= \left\{ (\mu, \nu) : |\mu| \leq \frac{1}{2}, \ \nu \leq -1 \right\}, \\ D_4 &= \left\{ (\mu, \nu) : |\mu| \geq \frac{1}{2}, \ \nu \leq -\frac{2}{3} (|\mu| + 1) \right\}, \\ D_5 &= \left\{ (\mu, \nu) : |\mu| \leq 2, \ \nu \geq 1 \right\}, \\ D_6 &= \left\{ (\mu, \nu) : 2 \leq |\mu| \leq 4, \ \nu \geq \frac{1}{12} (\mu^2 + 8) \right\}, \\ D_7 &= \left\{ (\mu, \nu) : |\mu| \geq 4, \ \nu \geq \frac{2}{3} (|\mu| - 1) \right\}, \\ D_8 &= \left\{ (\mu, \nu) : \frac{1}{2} \leq |\mu| \leq 2, \ -\frac{2}{3} (|\mu| + 1) \leq \nu \leq \frac{4}{27} (|\mu| + 1)^3 - (|\mu| + 1) \right\}, \\ D_9 &= \left\{ (\mu, \nu) : |\mu| \geq 2, \ -\frac{2}{3} (|\mu| + 1) \leq \nu \leq \frac{2|\mu| (|\mu| + 1)}{\mu^2 + 2|\mu| + 4} \right\}, \\ D_{10} &= \left\{ (\mu, \nu) : 2 \leq |\mu| \leq 4, \ \frac{2|\mu| (|\mu| + 1)}{\mu^2 + 2|\mu| + 4} \leq \nu \leq \frac{1}{12} (\mu^2 + 2) \right\}, \\ D_{11} &= \left\{ (\mu, \nu) : |\mu| \geq 4, \ \frac{2|\mu| (|\mu| + 1)}{\mu^2 + 2|\mu| + 4} \leq \nu \leq \frac{2|\mu| (|\mu| - 1)}{\mu^2 - 2|\mu| + 4} \right\}, \\ D_{12} &= \left\{ (\mu, \nu) : |\mu| \geq 4, \ \frac{2|\mu| (|\mu| - 1)}{\mu^2 - 2|\mu| + 4} \leq \nu \leq \frac{2}{3} (|\mu| - 1) \right\}. \end{split}$$ Moreover, all the inequalities are sharp.

Lemma 3.2. [10] Let $f \in \mathcal{B}_0$ be of the form (3.1). Then for $\lambda \in \mathbb{C}$ , we have $$|c_2 + \lambda c_1^2| \le \max\{1, |\lambda|\},$$ $|c_3 + (1 + \lambda)c_1c_2 + \lambda c_1^3| \le \max\{1, |\lambda|\}.$ and the inequalities are sharp.

Lemma 3.3. [26] Let f be in $S^*(\varphi)$ , where $\varphi(z) = 1 + \sum_{n=1}^{\infty} B_n z^n$ . Then the logarithmic coefficients $\gamma_n$ of f satisfy the following inequalities $$\sum_{n=1}^{\infty} n^2 |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} |B_n|^2,$$ $$\sum_{n=1}^{\infty} (n+1)^t |\gamma_n|^2 \le \frac{1}{4} \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} |B_n|^2, \quad t \le 2.$$ The first inequality is sharp for the function f given by $zf'(z)/f(z) = \varphi(z)$ . Remark 3.1. The authors in [26] proved Lemma 3.3 under the additional hypothesis that $\varphi$ is univalent and starlike with respect to $\varphi(0) = 1$ , has a positive real part in $\mathbb{D}$ , and $\varphi'(0) > 0$ . However, after close inspection, we observed that these conditions are not necessary (see [26, Theorems 2.2 and 2.3]). <span id="page-9-3"></span>Lemma 3.4. [\[15,](#page-17-20) Th.3.2i.] Let h be convex in D, with h(0) = 1. If q is the analytic solution of $$q(z) + \frac{zq'(z)}{q(z)} = h(z), \quad q(0) = 1$$ and if Re q(z) > 0, then q is univalent. If p ∈ H with p(0) = 1 satisfies $$p(z) + \frac{zp'(z)}{p(z)} \prec h(z),$$ then p ≺ q and q is the best dominant. <span id="page-9-2"></span>Lemma 3.5. [\[14\]](#page-17-21) Let β and γ be complex numbers with β ̸= 0, and let h(z) = a + h1z + · · · be regular in D. If Re[βh(z) + γ] > 0 in D then the solution of $$q(z) + \frac{zq'(z)}{\beta q(z) + \gamma} = h(z), \quad q(0) = a$$ is analytic in D. Further, the solution satisfies Re[βq(z) + γ] > 0 and is given by $$q(z) = \begin{cases} H'(z) \left( \beta \int_0^z H''(t) t^{-1} dt \right)^{-1} - \gamma/\beta & \text{if } a = 0, \\ z^{\gamma} [H(z)]^{\beta a} \left( \beta \int_0^z [H(t)]^{\beta a} t^{\gamma - 1} dt \right)^{-1} - \gamma/\beta & \text{if } a \neq 0, \end{cases}$$ where $$H(z) = \begin{cases} z \exp\left(\frac{\beta}{\gamma} \int_0^z \frac{h(t)}{t} dt\right) & \text{if } a = 0, \\ z \exp\left(\int_0^z \frac{h(t) - a}{at} dt\right) & \text{if } a \neq 0. \end{cases}$$ The Gaussian hypergeometric function F(a, b; c; z) = ²F1(a, b; c; z) is defined in D by the power series expansion $$F(a,b;c;z) = {}_{2}F_{1}(a,b;c;z) = \sum_{n=0}^{\infty} \frac{(a)_{n}(b)_{n}}{(c)_{n} n!} z^{n},$$ where (a)ⁿ denotes the Pochhammer symbol, i.e., (a)⁰ = 1 and $$(a)_n = a(a+1)\cdots(a+n-1), \quad n = 1, 2, \dots$$ Here a, b, and c are complex numbers with c /∈ −N⁰ := {0, −1, −2, . . . }. By definition, we have F(a, b; c; z) = F(b, a; c; z). For further properties of the hypergeometric function, we refer to the handbook-1965-stegun-1965-stegun [\[1\]](#page-17-22). Recently, Sugawa [\[27\]](#page-18-2) obtained specific conditions on the parameters a, b, c for which the ratio of two hypergeometric functions is not a convex function. <span id="page-9-1"></span>Lemma 3.6. [\[27\]](#page-18-2) Let F(a, b; c; z) be the hypergeometric function with parameters a, b, c such that a+b−1 < c < a+b+ 1/2 and (c−a)(c−b) > 0. Then the function F(a + 1, b; c; z)/F(a, b; c; z) is not convex in D.