Let B₁(μ,β) denote the class of functions f(z)= z + a₂z²+ h+ a_nz^m+… that are analytic in the unit disc Δ and satisfy the condition Ref′(z)(f(z)/z)^⧎-1 > β, zεΔ, for some ⧎>0 and β< 1. Denote by S*(0)for B₁(0,0). For μ andc such thatc > -μ, letF =I_gm,c(f) be defined by$$F(z) = \left[ {\frac{{\mu + c}}{{Z^c }}\int_0^z {f^\mu (t)} t^{c - 1} dt} \right]^{1/\mu } ,z \in \Delta .$$ The author considers the following two types of problems: (i) To find conditions on ⧎,c and ρ > 0 so thatfεB₁(μ -ρ) implies I_μ,c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB₁ (μ -δ) impliesI_μο(f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 0 in Δ then the nth partial sumf_n off satisfiesf_n(z)/z≺ -1 -(2/z)log(l -z)in Δ. Here, ≺ denotes the subordination of analytic functions with univalent analytic functions. As applications we also give few examples.

Convolution technique and subordination theorem are used to study certain class of meromorphic univalent functions in the punctured unit disc.

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r²) < (2−r²)F(a,b;a +b;r²) holds for all r∈(0,1).

In this paper we characterize John domains in terms of John domain decomposition property. In addition, we also show that a domain 𝐷 in $\mathbb{R}^n$ is a John domain if and only if $D\backslash P$ is a John domain, where 𝑃 is a subset of 𝐷 containing finitely many points of 𝐷. The best possibility and an application of the second result are also discussed.

Let $\mathcal{U}(\lambda, \mu)$ denote the class of all normalized analytic functions 𝑓 in the unit disk $|z| < 1$ satisfying the condition

\begin{equation*}\frac{f(z)}{z}\neq 0\quad\text{and}\quad\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu +1}-1\right| < \lambda,\quad |z| < 1.\end{equation*}

For $f\in\mathcal{U}(\lambda, \mu)$ with $\mu\leq 1$ and $0\neq\mu_1\leq 1$, and for a positive real-valued integrable function 𝜑 defined on [0,1] satisfying the normalized condition $\int^1_0\varphi(t)dt=1$, we consider the transform $G_\varphi f(z)$ defined by

\begin{equation*}G_\varphi f(z)=z\left[\int^1_0\varphi(t)\left(\frac{zt}{f(tz)}\right)^\mu dt\right]^{-1/\mu 1},\quad z\in\Delta.\end{equation*}

In this paper, we find conditions on the range of parameters 𝜆 and 𝜇 so that the transform $G_\varphi f$ is univalent or star-like. In addition, for a given univalent function of certain form, we provide a method of obtaining functions in the class $\mathcal{U}(\lambda, \mu)$.

The main results of this paper are characterizations of John disks–the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain’s hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property.

In this paper, we first establish a Schwarz–Pick type theorem for pluriharmonic mappings and then we apply it to discuss the equivalent norms on Lipschitz-type spaces. Finally, we obtain several Landau’s and Bloch’s type theorems for pluriharmonic mappings.

In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for a class of univalent harmonic functions which includes functions convex in some direction. Next, we prove growth and covering theorems and some related results. Finally, we propose two conjectures, an affirmative answer to one of which would then imply, for example, a solution to the conjecture of Clunie and Sheil-Small.