In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here.

For analytic functions f in the unit disk 𝔻 normalized by f(0)=0 and f'(0)=1 satisfying in 𝔻 the condition Re { (1-z)f'(z) }> 0, Re { (1-z^2)f'(z) }> 0, Re { (1-z+z^2)f'(z) }> 0, Re { (1-z)^2f'(z) } > 0, respectively, the upper bound of the third logarithmic coefficient was computed.

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for the tilted Caratheo dory function of order beta and angle alpha (alpha is an element of (-pi/2, pi/2) and beta is an element of [0, cos alpha)), which maps the unit disk D into the region {w is an element of C : Re{e(i alpha) w} > beta}. Using these conditions, we also derive conditions for an analytic function that maps D into a sector defined by {w is an element of C : | arg(w - gamma)|< (pi/2)delta}, where gamma is an element of [0,1) and delta is an element of (0, 1]. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D.

The aim of the present paper is to investigate several new criteria on a function to be strongly close-to-convexity,spirallikeness or starlikeness by using a new methodology.

We prove sharp bounds for Hankel determinants for starlike functions f with respect to symmetrical points, i.e., f given by f(z) = z + �& INFIN; n=2 anzn for z & ISIN; D satisfying zf & PRIME;(z) Re f(z) - f(-z) > 0, z & ISIN; D. We also give sharp upper and lower bounds when the coefficients of f are real.

AbstractLet f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$ , that is, functions satisfying $$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$ for $z\in \mathbb {D}$ , $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$ . We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$ , thus solving an open problem and presenting some new inequalities for coefficient differences.

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions f is an element of S given by f(z) = z + Sigma(infinity)(n=2) a(n)z(n) for D = {z is an element of C : vertical bar z vertical bar < 1} has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form H-2,H-2 (f) = a(2)a(4) - a(3)(2). A non-sharp bound for H-2,H-2 (f ) when f is an element of K(alpha), alpha is an element of [0, 1) consisting of convex functions of order alpha was found by Krishna and Ramreddy (Hankel determinant for starlike and convex functions of order alpha. Tbil Math J. 2012;5:65-76), and later improved by Thomas et al. (Univalent functions: a primer. Berlin: De Gruyter; 2018). In this paper, we give the sharp result. Moreover, we obtain sharp results for H-2,H-2 (f(-1)) for the inverse functions f(-1) when f is an element of K(alpha), and when f is an element of S* (alpha), the class of starlike functions of order alpha. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and f(-1) for the classes S* (alpha), K(alpha), S-beta* and K-beta, where S-beta*, and K-beta are, respectively, the classes of strongly starlike, and strongly convex functions of order beta.

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions with f(0)=0, and f′(0)=1. Let F be the inverse function of f, given by F(z)=ω+∑n=2∞Anωn for some |ω|≤r0(f). Let S*⊂S be the subset of starlike functions in D, and C the subset of convex functions in D. We show that −1≤|A3|−|A2|≤3 for f∈S, the upper bound being sharp, and sharp upper and lower bounds for |A3|−|A2| for the more important subclasses of S* and C, and for some related classes of Bazilevič functions.

Let f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ , and $${{\mathcal {S}}}$$ be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ for $$z\in {\mathbb {D}}$$ . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ , and $$0<\beta \le 1$$ . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ , where $$f^{-1}$$ is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ and $$|H_2(2)(f^{-1})|$$ for strongly convex functions.

Let $f$ be analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and ${\mathcal{S}}$ be the subclass of normalised univalent functions given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ for $z\in \mathbb{D}$. We give sharp upper and lower bounds for $|a_{3}|-|a_{2}|$ and other related functionals for the subclass ${\mathcal{F}}_{O}(\unicode[STIX]{x1D706})$ of Ozaki close-to-convex functions.

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H2,2{H_{2,2}} or H3,1{H_{3,1}} over different subclasses of 𝒮{\mathcal{S}}. Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯{\mathcal{T}} of typically real functions. The second object of our interest is 𝒦ℝ⁢(i){\mathcal{K}_{\mathbb{R}}(i)}, the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that $p(0)=1$ to satisfy $\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma $ or $| \arg \{p(z)-\gamma \} |<\delta $ for all $z\in \mathbb{D}$ , where $\beta \in (-\pi /2,\pi /2)$ , $\gamma \in [0,\cos \beta )$ , $\delta \in (0,1]$ and $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$ . The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in $\mathbb{D}$ .

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions given by f(z)=z+∑n=2∞anzn for z∈D. Let S*⊂S be the subset of starlike functions in D and C⊂S the subset of convex functions in D. We give sharp upper and lower bounds for |a3|−|a2| for some important subclasses of S* and C.

Let f ∈ A , the class of normalized analytic functions defined in the unit disk D , and be given by f ( z ) = z + ∑ n = 2 ∞ a n z n for z ∈ D . This paper presents a new approach to finding bounds for | a n | . As an application, we find the sharp bound for | a 5 | for the class B 1 ( α ) of Bazilevič functions when α ≥ 1 .

In this paper, the sharp bounds for the third Hermitian Toeplitz determinant over classes of functions convex in the direction of the imaginary axis and convex in the direction of the positive real axis are computed.

The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class of convex functions are computed.

For some subclasses of close-to-star functions the sharp upper and lower bounds of the second and third-order Hermitian Toeplitz determinants are computed.

Let $\psi_{\mu,\nu}(z)=(1-2\cos\nu e^{i\mu}z+e^{2i\mu}z^{2})^{-1}$, $\mu,\nu\in[0,2\pi)$ and p be an analytic mapping with $\operatorname{Re} p>0$ on the open unit disk. We consider the sense-preserving planar harmonic mappings $f=h+\overline{g}$, which are shears of the mapping $\int_{0}^{z} \psi_{\mu,\nu}(\xi) p(\xi)\,{d}\xi$ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations $g'/h'$ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

The object of the present paper is to investigate various conditions for Carathéodory functions in the open unit disk. Also we give some applications to univalent functions as special cases.

In this paper, the authors derive several sufficient conditions for a function to be the Caratheodory function in the unit disk D := {z is an element of C : vertical bar z vertical bar< 1}. More precisely, for given beta is an element of (-pi/2, pi/2), gamma is an element of [0; cos beta) and delta is an element of (0; pi/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e(-i beta)p(z)} > gamma or vertical bar arg{p(z) - gamma}vertical bar < delta for all z is an element of D by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness. (C) 2019 Mathematical Institute Slovak Academy of Sciences