This note is to show that if $f$ is a nonconstant entire function that shares two pairs of small functions ignoring multiplicities with its first derivative $f\mathrm{\text{'}}$, then there exists a close linear relationship between $f$ and $f\mathrm{\text{'}}$. This result is a generalization of some results obtained by Rubel and Yang, Mues and Steinmetz, Zheng and Wang, and Qiu. Moreover, examples are provided to show that the conditions in the result are sharp.

For an analytic univalent function $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disk, it is well-known that $\left|a_n\right|\le n$ for $n\ge 2$. But the inequality $\left|a_n\right|\le n$ does not imply the univalence of $f$. This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order $\alpha$ () for functions with prescribed coefficient bound on the analytic part.

There are many articles in the literature dealing with the first-order and the second-order differential subordination and superordination problems for analytic functions in the unit disk, but only a few articles are dealing with the above problems in the third-order case (see, e.g., Antonino and Miller (2011) and Ponnusamy et al. (1992)). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller in (2011). Let Ω be a set in the complex plane $\mathbb{C}$. Also let $\mathfrak{p}$ be analytic in the unit disk and suppose that $\psi :{\mathbb{C}}^4\times \mathbb{U}\to \mathbb{C}$. In this paper, we investigate the problem of determining properties of functions $\mathfrak{p}(z)$ that satisfy the following third-order differential superordination: $\mathrm{\Omega }\subset \left\{\psi \left(\mathfrak{p}\left(z\right),z{\mathfrak{p}}^{\prime }\left(z\right),z^2{\mathfrak{p}}^{\prime \prime }\left(z\right),z^3{\mathfrak{p}}^{\prime \prime \prime }\left(z\right);z\right):z\in \mathbb{U}\right\}$. As applications, we derive some third-order differential subordination and superordination results for meromorphically multivalent functions, which are defined by a family of convolution operators involving the Liu-Srivastava operator. The results are obtained by considering suitable classes of admissible functions.

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classes $C^{\mathrm{1}}(U)$, respectively, and $C^{\mathrm{2}}(U)$ to be univalent and to map $U$ onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the class $C^{\mathrm{1}}$ which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes $C^{\mathrm{1}}$ and $C^{\mathrm{2}}$ following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). Let $\mathrm{\Omega }$ be any set in the complex plane $\mathbb{C}$, let $p$ be a nonanalytic function in the unit disc $U$, $p\in C^{\mathrm{2}}(U),$ and let $\psi (r,s,t;z):{\mathbb{C}}^{\mathrm{3}}\times U\to \mathbb{C}$. In this paper, we consider the problem of determining properties of the function $p$, nonanalytic in the unit disc $U$, such that $p$ satisfies the differential subordination $\psi (p(z),Dp(z),D^{\mathrm{2}}p(z)-Dp(z);z)\subset \mathrm{\Omega }\Rightarrow p(U)\subset \mathrm{\Delta }$.

In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant $|a_2{a}_4-a_3^2|$ for functions belonging to the subclasses $S(\alpha ,\beta )$, $K(\alpha ,\beta )$, $S_s^{\ast }(\alpha ,\beta )$, and $K_s(\alpha ,\beta )$ of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

An analytic function $f$ defined on the open unit disk is biunivalent if the function $f$ and its inverse $f^{-1}$ are univalent in $𝔻$. Estimates for the initial coefficients of biunivalent functions $f$ are investigated when $f$ and $f^{-1}$, respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.

Sufficient conditions are obtained to ensure starlikeness of positive order for analytic functions defined in the open unit disk satisfying certain third-order differential inequalities. As a consequence, conditions for starlikeness of functions defined by integral operators are obtained. Connections are also made to earlier known results.

We present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transformation, has connections with three well-known surfaces: Enneper’s surface, the wavy plane, and the helicoid. Moreover, the identification process used to recognize the surfaces provides a connection to surfaces that give tight bounds on curvature estimates first studied in a 1988 work by Hengartner and Schober.

Let be a biharmonic mapping of the unit disk $𝔻$, where $g$ and $h$ are harmonic in $𝔻$. In this paper, the Landau-type theorems for biharmonic mappings of the form $L(F)$ are provided. Here $L$ represents the linear complex operator $L=\mathrm{(z}\partial /\partial \mathrm{z)}-\overline{\mathrm{(z}}\partial /\partial \overline{\mathrm{z)}}$ defined on the class of complex-valued $C^1$ functions in the plane. The results, presented in this paper, improve the related results of earlier authors.

The class 𝒰 $(\lambda ,\mu )$ of normalized analytic functions that satisfy for all z in the open unit disk is studied and sufficient conditions for an α-convex function to be in 𝒰 $(\lambda ,\mu )$ are given.

By means of the normal family theory, we estimate the growth order of meromorphic solutions of some algebraic differential equations and improve the related results of Barsegian et al. (2002). We also give some examples to show that our results occur in some special cases.

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out.