Coefficient Inequalities Research Articles

Certain classes of analytic functions with varying arguments

For the subclasses B α and b α of analytic functions satisfying Re{ (z) z } > α (0 ⩽ α < 1) and Re{′(z)} > α (0 ⩽ α < 1) , respectively, we prove various coefficient inequalities and distortion theorems, and determine the radius of convexity and starlikeness of such functions. We also consider an interesting application of the fractional calculus involving analytic functions belonging to these subclasses.

New classification of analytic functions with negative coefficients

New classification of analytic functions with negative coefficients is given by using the coefficients inequality, that is, new subclass A(p, n, Bk) of analytic functions with negative coefficient is defined. The object of the present paper is to prove various distortion theorems for functions in A(p, n, Bk), and for fractional calculus of functions belonging to A(p, n, Bk). Further, some properties of the class A(p, n, Bk) are shown.

Iteration of functions subordinate to schlicht functions

For a fixed schlicht function G using iterations we find necessary and sufficient conditions on the coefficients of a function f so that f is subordinate to G. We also show how these iterations may be used to obtain sharp coefficient inequalities for such functions and apply the results to the Krzyz conjecture for 1 ≤ n ≤ 4.

The Coefficients of Univalent Functions Related to Goodman's Conjecture

We consider certain coefficient inequalities for univalent functions suggested by Lyzzaik and Styer. These inequalities are related to a coefficient conjecture of Goodman concerning multivalent functions.

Dealing with unequal diffusivities among reactants in kinetic studies at rotating disk and ring-disk electrodes: The reduction of H 2O 2 as catalyzed by Co II(cyclam) and Fe(II)

The analysis of homogeneously catalyzed electrode reactions at rotating disk and ring-disk electrodes is complicated when the concentrations of substrate and catalyst are comparable because pseudo-first-order kinetics are not observed and differences in the diffusion coefficients of substrate and catalyst become important. The effects of these factors on the currents measured at rotating disk and ring-disk electrodes were analyzed by means of simulations utilizing an explicit version of the finite difference algorithm described in our previous paper (J. Electroanal. Chem., 215 (1986) 11) with the introduction of the appropriate homogeneous kinetic scheme. As an experimental example, the electroreduction of H 2O 2 as catalyzed by Co II(cyclam) was examined and rate constants were extracted for simulated working curves. Previously reported data for the same process with Fe 2+ as catalyst (Prater and Bard, J. Electrochem. Soc., 117 (1970) 1517; Skinner et al., ibid., 127 (1980) 315) were re-analyzed using the newer algorithm with appropriate consideration of the inequality of diffusion coefficients. The neglect of the inequality of diffusion coefficients in these earlier studies is shown to have been insignificant under the experimental conditions employed.

The role of the collecting duct in urinary concentration

In summary, the three major segments of the collecting duct subserve three different functions in the urinary concentrating mechanism. The main function of the cortical collecting tubule is to raise the fractional solute contribution and absolute concentration of urea in fluid that it delivers to the outer medullary collecting duct. The function of the outer medullary collecting duct is to raise further the absolute intraluminal urea concentration. Finally, the inner medullary collecting duct has two major functions in urinary concentration: first, it adds net urea to the papillary interstitium, and second, it allows the generation of maximally concentrated urine due to osmotic water equilibration. Indeed, the urine osmolality can rise to levels higher than the papillary interstitial osmolality as a consequence of inequalities of the reflection coefficients of urea and sodium chloride.

On coefficient inequalities in the class $\Sigma$

On coefficient inequalities in the class $\Sigma$

Generalization of certain subclasses of analytic functions

We introduce the subclass Tj(n, m, α) of analytic functions with negative coefficients by the operator Dn. Coefficient inequalities and distortion theorems of functions in Tj(n, m, α) are determind. Further, distortion theorems for fractional calculus of functions in Tj(n, m, α) are obtained.

Univalence Constraints on the Schwarz–Christoffel Parameters

The Area Theorem and coefficient inequalities for univalent functions are used to derive inequality constraints on the accessory parameters in the Schwarz–Christoffel formula for the conformal mapping of the unit disk onto the interior or onto the exterior of a polygon. The constraints restrict the choice of prevertices and are necessary for the univalence of the mapping. The conditions are explicit, easy to compute and are applicable to the numerical computation of the Schwarz–Christoffel map. The relative effectiveness of the constraints is illustrated by elementary examples.

A coefficient inequality for functions of positive real part with an application to multivalent functions

A coefficient inequality for functions of positive real part with an application to multivalent functions

Coefficient inequalities for a subclass of starlike functions

A function f ( z ) = z − ∑ ∞ n = 2 a n z n , a n ⩾ 0, analytic and univalent in the unit disk, is said to be in the family T ∗ (a, b) , a real and b ⩾ 0, if ¦( zf′ f ) − a¦ ⩽ b for all z in the unit disk. A complete characterization is found for T ∗ (a, b) when a ⩾ 1. Also, sharp coefficient bounds are determined for certain subclasses of T ∗ (a, b) when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family.

Coefficient inequalities for conformal mappings with homeomorphic continuation

Coefficient inequalities for conformal mappings with homeomorphic continuation

On new classes of analytic functions with negative coefficients

We introduce the classes of analytic functions with negative coefficients by using the nth order Ruscheweyh derivative. The object of the present paper is to show coefficient inequalities and some closure theorems for functions f(z) in . Further we consider the modified Hadamard product of functions fi(z) in .

Numerical methods for the solution of the rotating disc electrode system

Finite-difference and orthogonal collocation methods are applied to the solution of the rotating disc electrode system. Both methods reduce the ordinary differential equation system, describing the steady-state condition at the rotating disc, to a set of simultaneous equations, which are readily solved using simple matrix operations, and both may be generally applied to coupled kinetic and convective diffusion problems. Orthogonal collocation is found to be by far the more efficient of the two methods considered and its efficiency can be optimized by using a change of the space variable, chosen to best suit the concentration function of which the numerical solution is sought. The method is applied to a specific second-order kinetic problem, including inequality of diffusion coefficients, of relevance to the two-step electron transfer process in micellar solution.

An analogue of a hardy-littlewood-fej�r inequality for upper triangular trace class operators

Let H denote a separable, infinite-dimensional Hilbert space. We shall consider operators (that is, bounded linear transformations) on/-/. An operator T is said to be in the Schatten class Cp if T is compact and if the eigenvalues of (T'T) t/2 are in 1 p (0<p<oo); the norm in Cp is the I p norm of these eigenvalues. The class C 2 is called the Hilbert-Schmidt class, and C 1 is called the trace class. By C~o we denote the class of all compact operators. For information on these classes see, for example, [23], [4, Chap. XI], [8, Chap. III], [19], [13. Although there is an obvious analogy with the sequence spaces I p, there is a deeper analogy with the function spaces L p on the unit circle (however, the containment relations are reversed from the L p case, namely, Cp c Cr for p < r). This analogy serves as a source for new conjectures in operator theory. Motivated by this we establish an operator analgue of a coefficient inequality for the function space H 1. First we survey some known results in order to emphasize the nature and depth of the analogy. Fix an orthonormal basis {en} (n= l, 2, ...) in H, and let En =span{e I .... , en}. Each operator T on/-/has a matrix representation:

Coefficient inequalities and maximalization of some functionals for pairs of vector functions

Coefficient inequalities and maximalization of some functionals for pairs of vector functions

Inequality of eddy transfer coefficients for vertical transport of sensible and latent heats during advective inversions

Similarity of transport of water vapour and sensible heat was investigated within an advective inversion layer by measuring eddy fluxes together with gradients of temperature and humidity. The experimental site was a field of rice, grown under flood irrigation, which was situated in a semi-arid region. The fetch was about 300 m and local stabilities (z/L) over the rice ranged from 0 to 0.1.

Space-charge effects in flowing-afterglow data analysis for negative-ion/molecule reactions

In positive-ion/negative-ion plasmas, the space-charge fields (ambipolar fields) in the flow tube vary as the ion/molecule reaction proceeds if the value of the diffusion coefficient of the product ions is not equal to that for the reactant ions. The variation of the fields, however, has not been incorporated in conventional flowing-afterglow data analysis. In this report a mathematical model of the data analysis with a variable-field term is presented for negative-ion/molecule reactions, as examples of reactions in ion/ion plasmas. The model is applied to the reactions F − + CH 3Br → Br − + CH 3F and Cl − + CH 3Br → Br − + CH 3Cl, and then the errors in the rate constants arising from neglecting the variation of the fields are evaluated with the aid of a computer for the flowing-afterglow conditions: average velocity of helium buffer gas, 8000 cm s −1; pressure, 0.40 Torr; and temperature, 298 K. The magnitude of the error depends on the extent of the inequality of the diffusion coefficients between the reactant and the product ions, on the reaction length, and on the extent of progress of the reaction. For the F − + CH 3Br reaction, the error amounts to 15–30% when the reaction length is 70 cm.

Transport across homoporous and heteroporous membranes in nonideal, nondilute solutions. I. Inequality of reflection coefficients for volume flow and solute flow

The Kirkwood formulation of the Stefan-Maxwell equations is used to develop the transport equations for a membrane bounded by nonideal, nondilute solutions. The reflection coefficients for volume flow and solute flow are not equal but are related by a simple expression that depends on the concentration of the bounding solutions. The ratio of the two coefficients is independent of heteroporous membrane structure and the thickness of adjacent boundary layers. Experimental measurements of these reflection coefficients for sucrose transport across Cuprophan verify this relationship; this indicates that the Onsager reciprocal relation, which is assumed by the theory, holds for nonideal, nondilute solutions. The two reflection coefficients may be made operationally identical by a simple redefination of the osmotic driving force.

Inequality of direct and converse piezoelectric coefficients

Abstract It is demonstrated theoretically and experimentally that the direct and converse piezoelectric coefficients. ∂P/∂σ and ∂ε/∂E, are not equal as is generally believed. The correct relationship between these coefficients is derived, and it is shown that the experimentally measured quantities analogous to the piezoelectric coefficients are equal. Similar considerations apply to the relationship between the pyroelectric and electrocaloric coefficients.