Extremal Problems in the Class of Close-to-Convex Functions

The class K of normalized close-to-convex functions in D = {z: z < 1} has a parametric representation involving two Stieltjes integrals. Using a variational method due to G. M. Goluzin (2) for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for K. With these variational formulas, two general extremal problems within K are solved. The first problem is to maximize the functional Re F(logf'(z)) over K where F(w) is a given entire function and z a given point in D. A special case of this is the rotation theorem for K. The second problem solved is a general coefficient problem. Both problems are solved by characterizing the measures which appear in the integral representation for the extremal functions. The classes of convex univalent functions in D and functions whose derivative has a positive real part in D are proper subclasses of K. The methods used to solve the extremal problems in K can be used for these subclasses as well. Some of the results for the subclasses are known and are not presented here, even though the methods differ from those used previously. It should be mentioned that Goluzin originally used these methods to solve extremal problems of the first type mentioned above within the classes of starlike and typically real functions.

Let f (z )= D(F(z)) where D is a differential operator defined separately in every result. Let F be analytic and F(0 )= 0,F � (0 )= 1. We shall find out the disc in which operator D transforms some classes of analytic functions into the same. if and only if p1,p2 ∈ P(n,β) for z ∈ E. It is easy to see that P2(1,β) ≡ P(β) and P(1,0) ≡ P. The class Pk(1,0) ≡ Pk was introduced by Pinchuk (6), where he has generalized the concept of functions of bounded boundary rotation. It is worth mentioning that, for k > 2, functions in Pk need not be with the positive real part. It is easy to see that p ∈ Pk(n,β) if and only if there exists h ∈ Pk(n,0) such that p(z )=( 1 − β)h(z )+ β.

In this paper we survey some author’s results and developments relating the so-called Gel’fond-Leont’ev (G-L) operators of generalized integration and differentiation, classes of special functions (SF) of generalized hypergeometric type and the operators of generalized fractional calculus (GFC). The G-L operators have been introduced by Gel’fond-Leont’ev [9] in the classes of analytic functions in disks ΔR = {|z| < R}, by means of of multipliers’ sequences composed by the coeffcients of suitable entire (generating) functions. Introducing classes of SF related to Fractional Calculus (FC), as the Mittag-Leffer (ML) function, the multi-index Mittag-Leffer (multi-ML) function and its various particular cases ([16]–[18]), we specify the G-L operators generated by these entire functions. It is shown that in these cases, the G-L operators can be extended to analytic functions in wider complex domains Ω starlike with respect to the origin z = 0 and represented by operators of the Generalized Fractional Calculus (GFC), Kiryakova [14], i.e. operators of generalized integration and differentiation of arbitrary fractional multi-order. Illustrative examples and some open problems are proposed.

that converges uniformly interior to E. Consequently, if U(x, y) is any function that is harmonic in E, then U can be represented as the real part of some f that is given by (1). We shall use this fact to establish a Poisson integral formula for U. We should mention here that the Poisson formula can also be obtained by mapping the ellipse in the z-plane onto a rectangle in the w-plane thus obtaining F(w) as the transform of f(z) and applying the Cauchy integral formula, using in addition some properties of the Jacobian elliptic function Z(w). We have chosen the method presented in this paper because of its simplicity. The Poisson formula is readily generalized to the Poisson-Stieltjes formula and to what is commonly called the Herglotz formula for nonnegative harmonic functions. We apply the Poisson-Stieltjes formula to the class 61 of functions with positive real part in E and extend it to find an integral representation for functions regular in E and mapping E onto a domain that is starlike with respect to the origin. Other classes of functions related to 61 in various ways could also be studied by this method. The coefficient problem for the class of starlike functions as well as other subclasses of functions that are univalent in E and can be related to 61 is studied in [4].

Introduction.Let P denote the class of functions which are regular and satisfy Re /' (z) > 0 for | z \ < 1 and are normalized by /(0) = 0 and /' (0) = 1.This paper develops some properties of functions in P.An early consideration of functions satisfying the condition Re/'(z)>0 can be found in a paper by J. W. Alexander [l, p. 18].He proves: if f(z) is regular in \z\ <1 and f'(z) "maps \z\ <1 upon a region contained within a half-plane bounded by a straight line through the origin" then/(2) is schlicht for \z\ <1.J. Wolff [ll] showed that/(z) is schlicht in Re z>0 if it is regular there and satisfies Re/'(z)>0.K. Noshiro [6, p. 151 ] and S. Warschawski [10, p. 312] each demonstrated that Re/'(z)>0 is a sufficient condition for the schlichtness of f(z) in any convex domain.Conversely, S. R. Tims [9] proved that for each simply connected nonconvex domain D there is a function/(z) regular in D such that Re/'(z)>0 and f(z) is not schlicht in D. This result is a particular consequence of some more general theorems contained in a paper by F. Herzog and G. Piranian [2].They determine both necessary and sufficient conditions for a domain D-not necessarily simply connected-to have the property that every function regular and satisfying Re/'(z)>0 in D is schlicht there.A more general class of functions than those satisfying Re/'(z)>0 is the class of close-to-convex functions.W. Kaplan [3] calls a function f(z) close-to-convex in |z| <1 providing there is a function g(z) analytic, schlicht and convex in \z\ <1 for which Re{/'(z)/g'(z)} >0.Each function close-toconvex in \z\ <1 is schlicht there.Because of Alexander's result each function in P is schlicht in \z\ <1.Functions in Pean be constructed by integrating functions g(z) = I+&1Z+ which satisfy Reg(z)>0for \z\ <1.The function/(z) = -z -2 log (1-z) is in P for its derivative if f'(z) = (l+z)/(l-z) and w=(l+z)/(l-z) maps \z\ <1 onto Re w>0. 2. Distortion theorems.The following lemma contains results due to C. Caratheodory.A proof can be found in [7, Vol. 1, Problem 235, p. 129, and Vol. 1, Problem 287, p. 140].Lemma.If g(z) = l+ Sn-i onzn is regular in \z\ <1 and Re g(z)>0 then

Introduction.Let P denote the class of functions which are regular and satisfy Re /' (z) > 0 for | z \ < 1 and are normalized by /(0) = 0 and /' (0) = 1.This paper develops some properties of functions in P.An early consideration of functions satisfying the condition Re/'(z)>0 can be found in a paper by J. W. Alexander [l, p. 18].He proves: if f(z) is regular in \z\ <1 and f'(z) "maps \z\ <1 upon a region contained within a half-plane bounded by a straight line through the origin" then/(2) is schlicht for \z\ <1.J. Wolff [ll] showed that/(z) is schlicht in Re z>0 if it is regular there and satisfies Re/'(z)>0.K. Noshiro [6, p. 151 ] and S. Warschawski [10, p. 312] each demonstrated that Re/'(z)>0 is a sufficient condition for the schlichtness of f(z) in any convex domain.Conversely, S. R. Tims [9] proved that for each simply connected nonconvex domain D there is a function/(z) regular in D such that Re/'(z)>0 and f(z) is not schlicht in D. This result is a particular consequence of some more general theorems contained in a paper by F. Herzog and G. Piranian [2].They determine both necessary and sufficient conditions for a domain D-not necessarily simply connected-to have the property that every function regular and satisfying Re/'(z)>0 in D is schlicht there.A more general class of functions than those satisfying Re/'(z)>0 is the class of close-to-convex functions.W. Kaplan [3] calls a function f(z) close-to-convex in |z| <1 providing there is a function g(z) analytic, schlicht and convex in \z\ <1 for which Re{/'(z)/g'(z)} >0.Each function close-toconvex in \z\ <1 is schlicht there.Because of Alexander's result each function in P is schlicht in \z\ <1.Functions in Pean be constructed by integrating functions g(z) = I+&1Z+ • • • which satisfy Reg(z)>0for \z\ <1.The function/(z) = -z -2 log (1-z) is in P for its derivative if f'(z) = (l+z)/(l-z) and w=(l+z)/(l-z) maps \z\ <1 onto Re w>0. 2. Distortion theorems.The following lemma contains results due to C. Caratheodory.A proof can be found in [7, Vol. 1, Problem 235, p. 129, and Vol. 1, Problem 287, p. 140].Lemma.If g(z) = l+ Sn-i onzn is regular in \z\ <1 and Re g(z)>0 then

The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semi-definite real parts there and take selfadjoint operator values at real points, and its subclass consisting of functions representable in the form of Schur complement of a block of a linear pencil of operators with positive semidefinite operator coefficients, are investigated. The latter subclass is a generalization of the class of characteristic matrix functions of passive 2n-poles considered as functions of impedances of its elements, which was introduced by M. F. Bessmertnyi. Several equivalent characterizations of the generalized Bessmertnyi class are given, and its intimate connection with the Agler-Schur class of holomorphic contractive operator-valued functions on the unit polydisk is established.

We study the subclass Wσ(A) of the class of entire transcendental functions f(z)of exponential type with index not greater than σ satisfying the condition $$\int_{ - \infty }^\infty {\left| {f(x)} \right|^2 dx \leqslant A^2 .}$$ We find the set of values of the quantities f(z), f′(z), etc. when z is fixed and f runs through the subclass Wσ(A). We study extremal values of functionals of the type Φ(f(z), f ′(z)). In particular, we obtain upper bounds on the quantities ¦f(z +β/2) ± f(z−β/2) ¦ and ¦af '(z) + bof(z)¦.

In this note it is proved that entire functions of the form $$ F_a \left( z \right) = H\left( z \right) - a\alpha \left( z \right) $$ , where H and α are entire functions, with α having at least one zero, H′ and α′ having no common zeros and T(r, α) = S(r, H), are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.

In this paper, we introduce and study some new subclasses of analytic functions defined in the open unit disc using the convolution technique. Inclusion results, radius problems and several other properties of these classes are discussed. Key words: Convolution integral operator, functions with positive real part, alpha-starlike, bounded Mocanu variation, univalent.

The extreme points of a class of analytic functions with positive real part and a prescribed set of values

On the stability of certain systems under resonance conditions

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.

This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| &lt; 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).Our second theorem is a generalization of this. Namely, ifis analytic for |z| &lt; 1 and satisfies Re{zmf(m&gt;(z)} ≧ A andthenconverges.Another application of our fundamental inequality is the following. Let be analytic for |z| &lt; 1 and satisfy Re p(z) ≧ 0 and set and .

We consider the problem of stabilizing a matrix by a correction of minimal norm: Given a square matrix that has some eigenvalues with positive real part, find the nearest matrix having no eigenvalue with positive real part. It can be further required that the correction have a prescribed structure, e.g., be real, have a prescribed sparsity pattern, or have a given maximal rank. We propose and study a novel approach to this nonconvex and nonsmooth optimization problem, based on the solution of low-rank matrix differential equations. This enables us to compute locally optimal solutions in a fast way, also for higher-dimensional problems. Illustrative numerical experiments provide evidence of the efficiency of the method. It is further shown that the approach applies equally to the related problems of closed-loop stabilization of control systems and to the stabilization of gyroscopic systems.