Sharp Coefficient Estimates Research Articles

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

Sharp coefficient estimates for $\vartheta$-spirallike functions involving generalized q-integral operator

The aim of this article is to identify a new subfamily of spirallike functions and then to demonstrate necessary and sufficient conditions, sharp coefficients estimates for functions in this subfamily.

On Sharp Estimates of Coefficients for Polynomials in Norm-Attainable Classes

New sharp estimates for coefficients of pseudo-monoidal type polynomials are given for the class of norm-attainable operators. Moreover, we show the existence of an equivalence relation in the case of the class of norm-attainable operators with the Bergman determinant for the class of composition operators.

Certain Coefficient Estimate Problems for Three-Leaf-Type Starlike Functions

In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this class are addressed. Furthermore, we obtain the Fekete–Szegö inequality, sharp upper bounds for second and third Hankel determinants, bounds for logarithmic coefficients, and third-order Hankel determinants for two-fold and three-fold symmetric functions.

On the parametric representation of univalent functions on the polydisc

We consider support points of the class $S^0(\mathbb{D} ^n)$ of normalized univalent mappings on the polydisc $\mathbb{D} ^n$ with parametric representation, and we prove sharp estimates for coefficients of degree 2.

Sharp estimates for coefficients of odd trigonometric polynomials under a one-sided constraint

We study the largest and the smallest values of coefficients of odd trigonometric polynomials upper bounded by the function ϕ(x) = x on the interval [0, 2π]. A similar problem for the first and second coefficients was studied by the author earlier.

Harmonic K-quasiconformal Mappings from Unit Disk onto Half Planes

We consider the analytic properties for sense-preserving univalent harmonic mappings on the unit disk D onto half planes \(\Omega =\{w: \text {Re} w> a, -\infty 0, f_{\bar{z}}(0)\in R\). By deriving an analytic representing formula for these univalent harmonic mappings, we obtain one necessary and sufficient condition for these sense-preserving univalent harmonic mappings to be harmonic K-quasiconformal mappings, we also obtain sharp coefficient estimates for these harmonic K-quasiconformal mappings. As an application, distortion theorems on length and area for their images are also obtained. Our results improve and generalize the one made by M. Ozturk.

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings f ∈TL a , which are locally univalent harmonic map- pings in the unit disk D:={z:|z|<1}. Moreover, denoting the subclass TS a of the normalized univalent harmonic mappings, we also estimate the growth of |f|, f ∈TS a ,

Sharp coefficient estimates for certain subclasses of starlike functions of complex order

In this paper, the authors derive several sharp coefficient estimates for the subclassesS(λ,γ,A,B)andK(λ,γ,A,B,m;μ)of the class of normalized starlike functions of complex order γ∈C⧹{0}. The results presented here would generalize as well as improve the recent work by Srivastava et al. (2011) [20]. Some closely-related open problems are also posed.

Coefficient estimates for the inverses of a certain general class of spirallike functions

In the present paper, the authors derive several sharp coefficient estimates for the function class Sˆαβ-1 consisting of the inverses of functions in a certain class Sˆαβ of spirallike functions in the open unit disk U, which was introduced by Libera. They also obtain a number of sharp coefficient bounds of functions in the more general classes Σαβ and Σαβ-1, each of which is introduced here. Some of the results derived in this paper would generalize those in a recent work of Kapoor and Mishra.

Sharp coefficient estimates for a certain general class of Spirallike functions by means of differential subordination

In the present paper, the authors derive several sharp estimates for the Taylor-Maclaurin coefficients of functions in a certain general class S?(A, B) of spirallike functions in the open unit disk U, which is defined here by using the principle of differential subordination The results presented here would generalize those given in the earlier work of R. J. Libera.

On a subclass of close to convex functions

Let C′(α,β) be the class of functions f ( z ) = z + ∑ n = 2 ∞ a n z n analytic in D = { z : | z | < 1 } , satisfying for some convex function g( z) with g(0)= g′(0) − 1 = 0 and for all z in D the condition | z f ' ( z ) g ( z ) - 1 z f ' ( z ) g ( z ) + ( 1 - 2 α ) | < β for some α, β (0≤α<1, 0<β≤ 1). A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C′(α,β). The results extend the work of C. Selvaraj.

Certain subclasses of meromorphic harmonic starlike functions

In a previous paper M.-P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu and Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25–34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. In this paper, we employ this method to functions meromorphic harmonic in Ũ={z:|z|>1}. Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.

On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

The purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.

Coefficient estimates for inverses of starlike functions of positive order

In the present paper, the coefficient estimates are found for the class S ∗ − 1 ( α ) consisting of inverses of functions in the class of univalent starlike functions of order α in D = { z ∈ C : | z | < 1 } . These estimates extend the work of Krzyz, Libera and Zlotkiewicz [J.G. Krzyz, R.J. Libera, E. Zlotkiewicz, Coefficients of inverse of regular starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 33 (10) (1979) 103–109] who found sharp estimates on only first two coefficients for the functions in the class S ∗ − 1 ( α ) . The coefficient estimates are also found for the class Σ ∗ − 1 ( α ) , consisting of inverses of functions in the class Σ ∗ ( α ) of univalent starlike functions of order α in V = { z ∈ C : 1 < | z | < ∞ } . The open problem of finding sharp coefficient estimates for functions in the class Σ ∗ ( α ) stands completely settled in the present work by our method developed here.

On meromorphically harmonic starlike functions with respect to symmetric conjugate points

In a previous paper M.P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu, Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25–34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. Then, the same method is employed to functions meromorphic by Z.Z. Zou and Z.-R. Wu [Zhong Zhu Zou, Zhuo-Ren Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 261 (2001) 17–27]. Now, the method can be employed to functions meromorphic harmonic in the punctured disc 0 < | z | < 1 . Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.

Landau and Bloch theorems for harmonic mappings

Chen, Gauthier and Hengartner obtained some versions of Landau's theorem for bounded harmonic mappings and Bloch's theorem for harmonic mappings which are quasiregular and for those which are open. Later, Dorff and Nowak improved their estimates concerning Landau's theorem. In this study, we improve these last results by obtaining sharp coefficient estimates for properly normalized harmonic mappings. Furthermore, our estimates allow us to improve Bloch constant for open harmonic mappings.

Paley Type Inequalities for Orthogonal Series with Vector-Valued Coefficients

We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients with respect to a general orthonormal system Φ. This leads us to introduce the notions of Paley Φ-type and Φ-cotype for a Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.

On Functions α-Starlike with Respect to Symmetric Conjugate Points

We introduce here the notion of functions α-starlike with respect to symmetric conjugate points and derive a convolution theorem in this class. Moreover, a sharp coefficient estimate and a structural formula are given.

ON A CLASS OF MEROMORPHIC STARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS

Let $T^*_M(A, B, z_0)$ denote the class of functions \[f(z)=\frac{a}{z}-\sum_{n=1}^\infty a_nz^n, a\ge 1, a_n\ge 0\] regular and univalent in unit disc $U'=\{z:0&lt;|z|&lt;1\}$, satisfying the condition \[-z\frac{f'(z)}{f(z)}=\frac{1+Aw(z)}{1+Bw(z)}, \quad \text{ for } z\in U' \text{ and } w\in E\] (where $E$ is the class of analytic functions $w$ with $w(0) = 0$ and $|w(z)| \le 1$), where $-1\le A &lt; B \le 1$, $0\le B \le 1$ and $f(z_0) =1/z_0$ ($0&lt;z_0&lt;1$). In this paper sharp coefficient estimates, distortion properties and radius of meromorphic convexity for functions in $T^*_M(A, B, z_0)$ have been obtained. We also study integral transforms of functions in $T^*_M(A, B, z_0)$. In the last, it is proved that the class $T^*_M(A, B, z_0)$ is closed under convex linear combinations.