A differential inequality and starlikeness of a double integral

2015

Let Fp(b,M) denote the class of functions f(z) = z + ∑∞ k=2 akz k which are analytic in the open unit disc U = {z : |z| < 1} and satisfy the inequality ∣∣∣∣∣∣∣ b− 1 + zf ′ (z) f(z) b −M ∣∣∣∣∣∣∣ < M for b 6= 0, complex,M > 1 2 , |a2| = 2p, 0 ≤ p ≤ ( 1 +m 2 ) |b| , m = 1− 1 M and for all z ∈ U. Further f(z) is in the class Gp(b,M) if zf ′ (z) is in the class Fp(b,M). In the present paper, we obtain lower bounds for the classes introduced above and apply them to determine γ-spiral radiu for functions of the class Fp(b,M) and γ-convex radius for functions of the class Gp(b,M). 2010 Mathematics Subject Classification. 30C45.

Computers & Mathematics with Applications, 2011

In this paper we consider the classes of k-uniformly convex and k-starlike functions defined in Kanas and Wiśniowska (1999, 2000) [1,2] which generalize the class of uniformly convex functions introduced by Goodman (1991) [3]. We discuss the real part of f (z)/z, when f is k-starlike. We find the minimum of Ref (z)/z improving the results obtained recently in Wiśniowska-Wajnryb (2009) [11].

arXiv: Complex Variables, 2018

By using the method of differential subordination, we study a certain subclass of strongly starlike functions which is denoted by $\mathcal{S}^*_t(\alpha_1,\alpha_2)$, including of all normalized and analytic functions satisfying the following two--sided inequality: \begin{equation*} -\frac{\pi\alpha_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{\pi\alpha_2}{2} \quad |z|<1, \end{equation*} where $0<\alpha_1,\alpha_2\leq1$. The object of the present paper is to derive some certain inequalities for the desired class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$.

Thermal Science

Many mathematical concepts are explained when viewed through complex function theory. We are here basically concerned with the form f(z)=a0+a1z+a2z2+...f(z) ?A, f(z)=z+??, n=2anZ2 will be an analytic function in the open unit disc U={z:|z|<1, z=?C}normalized by f(0) = 0, f'(0)=1. In this work, starlike functions and close-to-convex functions with order 1/4 have been studied according to the exact analytic requirements.

Journal of Nonlinear Analysis and Application, 2011

Let A n be the class of analytic functions f of the form f (z) = z + ∞ ∑ k=n+1 a k z k z ∈ ∆, where n ∈ N is fixed. For λ > 0, α > 0 and µ > 0, we define a new class U n (α, λ, µ), of non-Bazilevič analytic functions by

International Journal of Mathematics and Mathematical Sciences, 2005

In 1999, Kanas and Rønning introduced the classes of starlike and convex functions, which are normalized with f (w) = f (w) − 1 = 0 and w a fixed point in U. In 2005, the authors introduced the classes of functions close to convex and α-convex, which are normalized in the same way. All these definitions are somewhat similar to the ones for the uniform-type functions and it is easy to see that for w = 0, the well-known classes of starlike, convex, close-to-convex, and α-convex functions are obtained. In this paper, we continue the investigation of the univalent functions normalized with f (w) = f (w) − 1 = 0, where w is a fixed point in U.

Demonstratio Mathematica, 1998

In this paper, some conditions have been improved so that the function g(z) is defined as g(z) = 1 + ∞ ∑ k≥2 a n+k z n+k , which is analytic in unit disk U , can be in more specific subclasses of the S class, which is the most fundamental type of univalent function. It is analyzed some characteristics of starlike and convex functions of order 2 −r. 2020 Mathematics Subject Classification. 30C45.

arXiv: Complex Variables, 2018

Let $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ denote the class of functions $f$ analytic in the open unit disc $\Delta$, normalized by the condition $f(0)=0=f'(0)-1$ and satisfying the following two--sided inequality: \begin{equation*} -\frac{\pi\alpha_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{\pi\alpha_2}{2} \quad (z\in\Delta), \end{equation*} where $0<\alpha_1,\alpha_2\leq1$. The class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ is a subclass of strongly starlike functions of order $\beta$ where $\beta=\max\{\alpha_1,\alpha_2\}$. The object of the present paper is to derive some certain inequalities including (for example), upper and lower bounds for ${\rm Re}\{zf'(z)/f(z)\}$, growth theorem, logarithmic coefficient estimates and coefficient estimates for functions $f$ belonging to the class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$.

2005

In the present investigation, we consider certain subclasses of starlike and convex functions of complex order, giving necessary and sufficient conditions for functions to belong to these classes.

In this paper, we obtained some properties for class of functions related to the class of starlike functions using a linear multiplier operator D n,q,s λ , f (z) (n ∈ N 0 , λ ≥ 0, ≥ 0), such as partial sums, integral means, square root and integral transform for these class are discussed.

TURKISH JOURNAL OF MATHEMATICS, 2019

Motivated by the Rønning-starlike class [Proceedings of the American Mathematical Society 1993; 118: 189-196], we introduce the new class S * c that includes analytic and normalized functions f , which satisfy the inequality Re zf ′ (z) f (z) ≥ f (z) z − 1 (|z| < 1). In this paper, we first give some examples that belong to the class S * c. Also, we show that if f ∈ S * c then Re{f (z)/z} > 1/2 in |z| < 1 (Marx-Strohhäcker problem). Afterwards, upper and lower bounds for |f (z)| are obtained where f belongs to the class S * c. We also prove that if f ∈ S * c and α ∈ [0, 1) , then f is starlike of order α in the disc |z| < (1 − α)/(2 − α). At the end, we estimate logarithmic coefficients, the initial coefficients, and the Fekete-Szegö problem for functions f ∈ S * c .

1977

Let M (a) denote the class of a-convex functions, a real, that is the class of analytic functions f (z) = z + E2 a" z" in the unit disc .D = {z: z < 1} which satisfies in D the condition f' (z) f (z)/z 0 and Re {(1_a) ' f'(z) + a 1 + ()} z f, (z) > 0. Let W (a) .f (z) ff z) denote the class of meromo-phic a-convex functions. a real, that is the class of analytic functions (z) = z + E, b" z' in D' _ {z: 0 < z < 1} which satisfies in D* the conditions zrAz)/^'.(z)-0 and Re ((l-a) (z^) + a f 1 +-z1` ;(z z) ^ G 0. In this paper we obtain the relation bctwcen M (a) and W (a). The radius of a-convexity for certain classes of starlike functions is also obtained.

2011

For $0\leq \alpha <1$, the sharp radii of starlikeness and convexity of order $\alpha$ for functions of the form $f(z)=z+a_2z^2+a_3z^3+...$ whose Taylor coefficients $a_n$ satisfy the conditions $|a_2|=2b$, $0\leq b\leq 1$, and $|a_n|\leq n $, $M$ or $M/n$ ($M>0$) for $n\geq 3$ are obtained. Also a class of functions related to Carath\'eodory functions is considered.

2011

In this paper, using the generalized Al-Oboudi operator we introduce the new subclass H μ λ (α, β) of analytic functions. We also consider the subclass H μ λ (α, β) of H μ λ (α, β). Coefficient inequalities and convolution conditions are investigated for these classes.