For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let ${}_{\gamma ,\beta }$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $Re\left(e^{i\gamma }P\left(z\right)\right)>\beta cos\gamma$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{}(z₀,\lambda )$ for ${\int }_0^{z₀}P\left(\zeta \right)d\zeta$ when P ranges over the class $\left(\lambda \right)=P{\in }_{\gamma ,\beta }:P^{\text{'}}\left(0\right)=2(1-\beta )\lambda e^{-i\gamma }cos\gamma .$As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

We consider functions of the type $F\left(z\right)=z{\prod }_{j=1}^n{[f_j\left(z\right)/z]}^{a_j}$, where $a_j$ are real numbers and $f_j$ are ${\beta }_j$-strongly close-to-starlike functions of order ${\alpha }_j$. We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

In the present work, we introduce the subclass ${𝒯}_{\gamma ,\alpha }^k\left(\varphi \right)$, of starlike functions with respect to $k$-symmetric points of complex order $\gamma$ ($\gamma \ne 0$) in the open unit disc $$. Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

The purpose of this paper is to study the class ${*}_{\infty }\left(ℍ\right)$ of univalent analytic functions defined in the right halfplane ℍ and starlike w.r.t. the boundary point at infinity. An analytic characterization of functions in ${*}_{\infty }\left(ℍ\right)$ is presented.

CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

We introduce and study two certain classes of holomorphic and bi-univalent functions associating $\lambda$-pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions belonging to these classes. Further we point out certain special cases for our results.

Let $\parallel ·\parallel$ be the uniform norm in the unit disk. We study the quantities $M_n\left(\alpha \right):=inf(\parallel zP\left(z\right)+\alpha \parallel -\alpha )$ where the infimum is taken over all polynomials $P$ of degree $n-1$ with $\parallel P\left(z\right)\parallel =1$ and $\alpha >0$. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{\alpha >0}{M}_n\left(\alpha \right)=1/n$. We find the exact values of $M_n\left(\alpha \right)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Let ${𝒫}_n$ denote the class of analytic functions $p\left(z\right)$ of the form $p\left(z\right)=1+c_n{z}^n+c_{n+1}{z}^{n+1}+\cdots$ in the open unit disc $𝕌$. Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p\left(z\right)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f\left(0\right)=f^{{}^{\text{'}}}\left(0\right)-1=0$. We introduce a subclass $S_M^{*}(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_M^{*}(n,\alpha ,b)$ ($n\ge 0$) related to $S_M^{*}(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_M^{*}(\alpha ,b)$. Several properties of the class $S_M^{*}(n,\alpha ,b)$ are investigated. ...

We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Let $P_k$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_k$ in $f={\sum }_k{a}_k{P}_k$. A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_k$ results in obtaining a low order of the recurrence.

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results...

The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $\alpha$ and from the class $\mathrm{𝒞𝒱\mathscr{H}}\left(\beta \right)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $\mathrm{𝒞𝒱\mathscr{H}}\left(\beta \right)$.

Let K be any subset of ${ℂ}^N$. We define a pluricomplex Green’s function $V_{K,\theta }$ for θ-incomplete polynomials. We establish properties of $V_{K,\theta }$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of ${ℝ}^N$, we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K\setminus supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$ when K is a compact...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

In this paper we investigate the factorization of the polynomials $f\left(x\right)x^n+g\left(x\right)\in \mathbb{Z}\left[x\right]$ in the special case where $f\left(x\right)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f\left(x\right)$ is monic and linear.

The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class $R^{\lambda }(a,c,A,B)$ of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ${\tilde{R}}^{\lambda }(a,c,A,B)$ of $R^{\lambda }(a,c,A,B)$ and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

Using the operator ${𝔇}_{q,\varrho }^m(\lambda ,l)$, we introduce the subclasses ${𝔜}_{q,\varrho }^{*m}(l,\lambda ,\gamma )$ and ${𝔎}_{q,\varrho }^{*m}(l,\lambda ,\gamma )$ of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.

We consider the class ${\mathscr{H}}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $\left|z\right|<1$ and normalized so that $h\left(0\right)=0=h^{\text{'}}\left(0\right)-1$ and $g\left(0\right)=0=g^{\text{'}}\left(0\right)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes ${𝒫}_H^0\left(\alpha \right)$ and ${𝒢}_H^0\left(\beta \right)$ of functions from ${\mathscr{H}}_0$ and show that if $f\in {𝒫}_H^0\left(\alpha \right)$ and $F\in {𝒢}_H^0\left(\beta \right)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections...