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The studies in operator theory are attracting many researchers. The central aim of this investigation is to formulate a special parametric differential operator (PDO) based on the error function in the open unit disk. The suggested operator is related to the well-known Carlson–Shaffer operator and the differential operator due to Salagean. We express a class of analytic functions to study its performance in view of the geometric function theory. As application, we employ PDO to present the conformable d'Alembert's equation. We discuss the univalent solution of the suggested equation and study some of its geometric behaviours.

The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known Wright f...

This paper concerns univalent analytic functions f for which log⁡f′ belong to Dirichlet-Morrey spaces. We establish some characterizations of Dirichlet-Morrey domains. Our results involve Schwarzia...

In current realization, our object is to use the convolution product in terms of the notion quantum calculus to deliver a propagated q-derivative factor taking a more generalized Sàlàgean formula. By joining both the new factor together with the Janowski formula, we designate a special category of analytic factors in domain of unit disk. Finally, we deliberate a set of significant inequalities involving these classes. As applications, we seek the q-differential translator to generalize a denomination of differential equations species Briot-Bouquet and formulate its upper analytic solution using the subordination idea. This application can be employed in information theory and thermo dynamical systems.

For , let denote the class of sense preserving harmonic mappings in the unit disk satisfying . The main aim of this paper is to study some basic properties such as coefficient bounds, growt...

This article studies extensions of the well-known Briot–Bouquet differential subordination resultWe extend the result by determining conditions on the functions and so thatThis extension is used to prove several new results about Alexander transforms of spiral-like functions, including a conjecture of Kim and Srivastava.

In this work, the Wright function with their normalizations are considered. Several conditions are obtained so that the Wright function has certain geometric properties including univalency, starlikeness, convexity and close-to-convexity in the open unit disk. Results obtained are new and their usefulness are depicted by deducing several interesting corollaries and relevance with some of the earlier results are also pointed out.

Let A{script} be the class of analytic functions in the unit disc with the normalization f (0) = f ?(0)-1 = 0. This article analyses various necessary and sufficient coefficient conditions for functions f ? A of the form to be univalent. We present an interesting class of univalent functions associated with the zeta function and also pose an open problem. � 2013 Copyright Taylor and Francis Group, LLC.

We establish the underlying quasiconformal features of the generalized Grunsky coefficients and provide their applications to qualitative and quantitative investigations of the complex holomorphic isotopies.

In this article we are concerned with some extension operators Φ n,α and Φ n,Q that provide a way of extending a locally univalent function f on the unit disc U to a locally biholomorphic mapping F ∈ H(B n ). By using the method of Loewner chains, we prove that if f can be embedded as the first element of a g-Loewner chain on the unit disc, where for |ζ

Let be a harmonic function in the unit disc, where f and g are holomorphic functions and g′(z) = zf′(z). We obtain a simple sufficient condition on f for the harmonic function h to be univalent and its image to be a three-cornered hat domain.

Every non-vanishing univalent function f (z) in the exterior of the unit disk Δ, with hydrodynamical normalization, generates the canonical isotopy f(z, r) = rf(r −1 z) to identity, which is used in many subjects. Let q f denote the Teichmuller norm of f. In 1995, Kuhnau posed several interesting questions on the properties of the dilatation function q f (r) = q f r , including the following: are the functions q f 1 (r) and q f 2 (r) different if f 1 ≠ f 2? We establish a strong rigidity for admissible dilatations q f (r), solving the question negatively.

In Lewandowski, Miller and Zlotkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc., 56 (1976), pp. 111–117 and Pascu, Alpha-close-to-convex functions, Romanian Finish Seminar on Complex Analysis, Springer, Berlin, 1979, pp. 331–335 it has been proved that the integral operator defined by Bernardi in Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), pp. 429–446 given by preserves certain classes of univalent functions such as the class of starlike functions, the class of convex functions and the class of close-to-convex functions. In this article, we determine conditions that a function f ∈ A needs to satisfy such that the function F given by (1) is starlike.

The purpose of the present article is to establish two theorems for a class of analytic univalent functions defined in the open unit disk. The results are proved by using techniques involving the principle of differential subordination. Connections of these theorems with Bazilevic˘ functions are considered and the main results are applied to obtain sufficient conditions for analytic functions to be starlike and close-to-convex.

Let U={z:| z

Conformal maps of the unit disk onto convex domains are a classical topic. Recently Avkhadiev and Wirths discovered that conformal maps onto concave domains (the complements of convex closed sets) have some novel properties, namely that there are non-trivial lower estimates, a rare thing for univalent functions (Avkhadiev, F.G. and Wirths, K.-J., 2002, Convex holes produce lower bounds for coefficients, Complex Variables, Theory and Application, 47, 556–563). We prove an inequality involving two variables and introduce a lower order for concave functions. §Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

We study the range of parameters λ and μ such that any function f (z), analytic for |z

In this article, we investigate the mappings of various subclasses of univalent functions by means of certain families of integral operators. Especially, by using the norm estimates of the pre-Schwarzian derivatives, we determine conditions under which these integral operators have geometric meaning associated with the typical subclasses of analytic and univalent functions.

Let ℋα (β) denote the class of normalized functions f, analytic in the unit disc E, which satisfy the condition: where α and β are pre-assigned real numbers. Al-Amiri and Reade, in 1975, showed that, for α ≤ 0 and also for α = 1, the functions in ℋα (0) are univalent in E. In the present article, among other things, we prove: (i) for 0 1/2 in E.

In this paper, some significant scientific errors in the paper by J.K. Prajapat entitled ‘Certain geometric properties of the Wright function’ are corrected.

Let H(U) be the space of all analytic functions in the unit disk U. For a function h ∈ A ⊂ H(U) we will determine simple sufficient conditions on h such that the integral operators Ĩ h; β, γ : [Ktilde] h; β, γ → H(U) defined by map a special subclass of analytic functions in 1/β-convex functions. Then, using a recent result of [T. Bulboaca. Generalization of a class of nonlinear averaging integral operators. Math. Nachr. (to appear).], we determined conditions on λ ∈ ℂ such that λI h; 1, γ[ƒ](z) ≺ ƒ(z) for all ƒ convex functions in U, where I h; β, γ 𝒦 h; β, γ → H(U) is given by In addition, some particular cases of this last result obtained for appropriate choices of the h function will be given.

Let Φ be a bounded positive continuous function in the complex plane C. We will consider the following problem. We can find a simply connected region Ω containing the origin such that the Riemann mapping function f Ω mapping the unit disk D onto Ω will satisfy We show that there is a unique solution which is starlike with respect to the origin when Φ satisfies We also show that if in addtion, ∂Φ/∂|w|>0 and −log Φ is subharmonic, then Ω will be a convex region.

We study bounded univalent functions ƒ(z) that map the unit disk into itself such that ƒ(0) = 0 and the angular limit (whose absolute value is equal to one) and the angular derivative exist at certain points of the unit circle. For such functions we obtain the sharp lower boundary curve for the range of the system of functionals (|ƒ(r)|,ƒ′(r)|). We use the method of moduli of curve families and the extremal partition of domains. Recently appeared entity of digon is used as well as symmetrization specified for digons by A. Solynin. In particular, we find that for r close to 0 the canonical Pick function gives the minimum to |ƒ′(r)| whereas for r close to 1 it is not the case.

Let B be the unit ball in C n and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ n,α that provide a way to extend a locally univalent function ƒ ∊ H(U) to a locally univalent mapping Fα ∊ H(B), where α ∊ (0,1]. If ƒ is normalized univalent, then Fα can be imbedded in a Loewner chain. Also if ƒ ∊ S∗, then Fα is starlike. We show that if ƒ belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping Fα satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C n (for n ≤ 2.)

The classical Bieberbach coefficient estimate and Koebe 1/4-theorem for univalent functions in the unit disk may be formulated, in terms of a naturally defined Bieberbach constant n(D) and Koebe constant C(D), as n(D)≤2 and C(D) ≥ 1/4 for each simply connected domain D. We show that these inequalities have the same extremal domains, by means of a variation on the classical argument that yields the Bieberbach and Koebe Theorems, and describe how this is related to work of Flinn and Herron and of Pommerenke. Such extremal domains satisfy n(D)C(D)=1/2. An example of a simply connected domain for which n(o:D)C(D)≠1/2 is constructed, thereby proving that equality does not always hold in the inequality n(D)C(D) ≥½, due to Osgood. In passing, we raise an interesting question on the relationship between the second coefficient of a univalent function f and the radius of the largest disk about f(0) that is covered by f

Let denote the family of functions f, normalized by f(0)=0=f′−1, that are analytic in the open unit disc and such that for some with . The main object of this paper is to study this class and to find conditions on α, β and on the function g such that each function f in belongs to a family which is contained in the family of univalent functions in the unit disc δ. We also find the exact value of in the class for fixed z∊δ. Further, we also determine condition on λ for functions f in P(λ) to be in the class of strongly starlike functions, or in the class of functions whose derivative lies in a sector of angle less than or equal to πγ/2 with γ∊(0,1]. Finally, we also obtain a sufficient condition for an analytic function f to satisfy the analytic univalence criteria of Noshiro-Warschawski. Several examples are stated in support of the sharpness of our criteria.

Let C 1(β) be the class of normalized functions f which are analytic in the open unit disk u, given by the power series: and satisfy the inequality: for some normalized univalent and convex function φ. In this paper we solve the Fekete-Szego problem for the family: by proving that

We prove that max , in the class S(M) of holomorphic, univalent and bounded functions in the unit disk |z| < 1 is realized by the Pick function, when M is close to 1.

Let αe[0,1], φ be an analytic function in a certain domain D in the planep be an analytic function in the unit disk U with p(U)⊂D and h be a convex function with 0eh(U). In this paper there is studied the differential subordination of the form , where zeU. The applications to the theory of univalent functions are presented.

Let f = h + g be a harmonic, univalent and orientation-preserving function on the unit disk, where h and g are analytic. We show that log|h(eit)| and log|g(eit)| are BMO when some Bloch conditions are satisfied.

LetAbe the class of functions analytic in the unit disk and normalized by f(0)=f′(0)+1=0 Let S, S∗(γ), and K(γ)be respectively the classes of normalized univalent functions, starlike functions of order γ and convex functions of order γ. In this paper we investigate the properties of the integral transform where λ is a non-negative real valued function normalized by ∫1 0 λ(t)dt=1. From our main results we get conditions on λ and the classFso that V λ (f) mapsF into various subclasses of the class of univalent functions. As a corollary to our results, we give an affirmative answer in support of a conjecture of Kim: f is a member ofSor S∗(γ) or K(γ), then the function φ(3,3+ αz) ∗ f(z) belongs to the same class for α >1, where φ(b,c;z) ∗ f(z) stands for the convolution of incomplete beta function with f∊ A

We investigate whether the conditions sufficient for a function that is analytic and locally univalent to be univalent in the unit disk Ucan be weakened by allowing the above three expressions to lie in a larger domain of variability. It turns out that in various cases the function f can then be infinitely valent in U.

A holomorphic function f defined on the unit disk is called strongly close-to-convex of order α > 0 if there is a convex univalent function φ defined on such that For α e (0.1) this condition f is univalent in . By convention a convex univalent function is called close-to-convex of order 0. If f is close-to-convex of order α e [0, 1 ] and , then where and is the hyperbolic distance between a and b. Equality holds for distinct if and only if where S is a conformal automorphism of automorphism , T is a conformal automorphism of ; and .

Let be a harmonic univalent and orientation-preserving function on the unit disk, where h and g are analytic. We first give an example to show that there can be no upper-bound on the valency of h. Then we show that h is Bloch if, and only if, h is BMOA if, and only if, f is BMOA and we show that log h is Bloch if, and only if, log h is B MOA. Finally, we give sharp estimates for the Bloch-norm and BMOH2-norm of f when f maps the unit disk onto the unit disk.

We characterize the open subsets G of the complex plane and the sequences of automorphisms of G for which there exists a universal holomorphic function f with the following property: the sequence is dense in the space of the holomorphic functions on G.

Let g be some set of non-constant analytic functions in the unit disk D, and for f e dg define The quantity where g u is the set of univalent functions in g, is called the John constant of g. We discuss where T k consists of the trinomials In particular, we find We also establish a conjecture of Rahman and Szynal for univalent trinomials in T k

Over the class S of normalized univalent functions in the unit disk, a variational method is used to find the maximum of |f(ζ|/R f for fixed ζ in the unit disk and where R f is the logarithmic capacity of the range of f. The solution is given explicitly in terms of standard elliptic integrals and implications for other work are described.

Let α e [0,1]p be an analytic function in the unit disc E q and h be analytic and univalent functions in E, Φ be an analytic function in a certain domain D⊃ q (E). The authors study the differential subordinations and where z⊂E Examples of the application of these results are given.

The object of the present paper is to prove various distortion theorems for fractional calculus of functions in the class Qn(α) consisting of analytic and univalent functions with negative coefficients in the unit disc. Furthermore, a distortion theorem for a fractonal integral operator of functions in the class Qn(α) is shown.

It is shown that the Pick functions are extremal for the functional |a 2a3| in the class of bounded univalent functions f(z)=z+ a 2z2+ …, |f(z)|< M,|z

A variational method is used to obtain the sharp upper bound for the second coefficients of univalent functions which map onto regions of prescribed logarithmic capacity (Transfinite diameter).

Let S be the class of normalized univalent functions. For and denote the class of such functions by IS. We investigate the range of the functionals F(z)/z and The results that we obtain are not sharp, but fairly good for small values of r. In particular we prove that

We prove that if f maps the unit disk D onto a convex domain with F(0)=0, then with equality holding for each f that has the prupeiiy that the left and right-nana tangents at each point w∊∂(D) are tangent to the circle This verifies the conjecture which we made in the early 1980's, which arose of our work on omitted value problems for classes of univalent functions. This conjecture was announced at the Purdue Symposium on the Occasion of the Proof of the Bieberbach Conjecture in 1985 and published in the article “Open Problems and Conjectures in Complex Analysis” (Computational Methods and Function Theory Lecture Notes in Math. #1435, 1990 pp. 1-26). The proof employs the Julia variation and properties of convexity and subordination. Also, we obtain, using this integral means estimate, a sharp bound for the convex case of Brennan's conjecture on the integral means with respect to area measure for the derivative of a univalent function.

The Koebe function k(z)=z(l−z)-2 and its rotations are solutions to many extremal problems over the class S of univalent functions. It is conjectured that if a function maximizes the real parts of two essentially different linear functionals over S, then it must be a rotation of the Koebe function. This paper considers coefficient functionals and gives some sufficient conditions for the conjecture to be true.

By the method of extremal length, three general theorems are proven which find, as corollaries, sharp coefficient bounds for functions univalent in the exterior of the unit disc, with a standard normalization but also assuming a finite number of initial coefficients are zero, which possess a K-quasiconformal extension to a ring subdomain of the unit disc. All extremal functions are exhibited. Similar estimates are found for functions similar to the above but which omit a fixed disc centered at the origin.

A classical theorem of Szego states that for functions univalent in |z

This paper continues carlier studies of the authors on the famulies f of lunction detined by where μ is a Borel measure on the unit circle and x

We prove an inequality between the area of the omitted set of an analytic function f and its coefficients. We show that equality holds if and only if f is a univalent function.

Let ƒ be a univalent function in the unit disk D with a quasiconformal extension such that the complex dilatation tends to zero near the unit circle We compare the complex dilatation with a related...

Let S be the class of functions f(z) = z + c 2 z 2 + … analytic and univalent in the disk Let be for f(z) ∈ S. One considers means of the form The choice x k which realizes extremal properties in the class S of the function is of main interest, especially the case x k = n - k + 1 (k= 1,…,), which corresponds to Milin's conjecture: In 1984 L. de Branges proved this conjecture. It is known that the famous Bieberbach conjecture, Robertson's conjecture and some other conjectures for the coefficients of univalent functions hold if I n is nonpositive for all n. In this paper we use the properties ofI n to obtain some inequalities for the functions of class S. In these inequalities the equality is attained by the function K x( z).