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In the paper we define classes of harmonic starlike functions with respect to symmetric points and obtain some analytic conditions for these classes of functions. Some results connected to subordination properties, coefficient estimates, integral representation, and distortion theorems are also obtained. [ABSTRACT FROM AUTHOR]

The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427–1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. However, the correct bound for the latter is stated and proved. Further, a sharp bound on the initial coefficients for normalized analytic function f such that z f ′ (z) / f (z) ≺ 1 + λ z , λ ∈ (0 , 1 ] , have also been obtained, which contain many existing results. [ABSTRACT FROM AUTHOR]

The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with the Bell numbers are determined. [ABSTRACT FROM AUTHOR]

The object of the present paper is to investigate various conditions for Carathéodory functions in the open unit disk. Also we give some applications to univalent functions as special cases. [ABSTRACT FROM AUTHOR]

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made. [ABSTRACT FROM AUTHOR]

Let g be an analytic function with the normalization in the open unit disk. Let L (r) be the length of g ({ z : | z | = r }) . In this paper we present a correspondence between g and L (r) for the case when g is not necessary univalent. Furthermore, some other results related to the length of analytic functions are also discussed. [ABSTRACT FROM AUTHOR]

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions. [ABSTRACT FROM AUTHOR]

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. [ABSTRACT FROM AUTHOR]

Let ψ μ , ν (z) = (1 − 2 cos ν e i μ z + e 2 i μ z 2) − 1 , μ , ν ∈ [ 0 , 2 π) and p be an analytic mapping with Re p > 0 on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g ‾ , which are shears of the mapping ∫ 0 z ψ μ , ν (ξ) p (ξ) d ξ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g ′ / h ′ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ. [ABSTRACT FROM AUTHOR]

In this study, a novel family of analytical functions connected to convex functions in the open unit is introduced and investigated. Additionally, relationships between this class and other subclasses of analytic functions are deduced. Further, different results for the mentioned class and several new interesting properties are obtained. [ABSTRACT FROM AUTHOR]

Due to the major importance of the study of the logarithmic coefficients for univalent functions, in this paper we find the sharp upper bounds for some expressions associated with logarithmic coefficients of functions that belong to some well-known classes of analytic functions in the open unit disk of the complex plane. [ABSTRACT FROM AUTHOR]

In this paper, we consider some relations related to the representations of starlike and convex functions, and obtain some sufficient conditions for starlike and convex functions by using the theory of differential subordination. Actually, we generalize a result by Suffridge for analytic functions with missing coefficients and then we apply that generalization for obtaining the different methods to the implications of starlike or convex functions. Our results generalize and improve the previous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the sharp and accurate bounds for the logarithmic coefficients of some subclasses of analytic functions defined and studied in earlier works. Furthermore, we obtain the bounds of the second Hankel determinant of logarithmic coefficients for a class defined by subordination, such as the class of starlike functions  S * (φ) . Some applications of our results, which are extensions of those reported in earlier papers are given here as special cases. In addition, the results given can be used for other popular subclasses. [ABSTRACT FROM AUTHOR]

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z 1 F 2 (a ; b , c ; z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F 1 0 (− ; c ; z) function are also mentioned. We further consider a boundedness property of the function F 2 1 (a ; b , c ; z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

In this work, we estimate the bounds for the logarithmic coefficients γ n of some well-known classes like F (c) for c ∈ (0 , 0.656 ] ∪ { 2 } . The best bound obtained for the logarithmic coefficient | γ 5 | is sharp for c = 2 . In a special case, it is important to note that we obtain the bounds for the logarithmic coefficients γ n of the convex functions of order α for some α . It is worthwhile mentioning that the given bounds would generalize some of the previous papers. [ABSTRACT FROM AUTHOR]

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U , then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0 , 1 and C V hpl 1 / 2 were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ C 1 + α z , then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1 , n ∈ ℕ. Equality is attained for the function L α , n , that is, log L α , n z / z = 2 ∑ n = 1 ∞ γ n L α , n z n = α / n n + 1 z n + ⋯ , z ∈ U. [ABSTRACT FROM AUTHOR]