QFT, EFT and GFT

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

Positivity and geometric function theory constraints on pion scattering

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with O(N) global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the z-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for O(N) model in z-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the O(N) model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, π+π−→ π0π0) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

Initial Coefficient Estimates and Fekete–Szegö Inequalities for New Families of Bi-Univalent Functions Governed by (p − q)-Wanas Operator

The motivation of the present article is to define the (p−q)-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions AE(λ,σ,δ,s,t,p,q;ϑ) and SE(μ,γ,σ,δ,s,t,p,q;ϑ) which are defined in the unit disk U associated with the (p−q)-Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters.

A new analytic solution of complex Langevin differential equations

PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

Fuzzy Differential Subordinations Obtained Using a Hypergeometric Integral Operator

This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. Using the ideas specific to geometric function theory from the field of complex analysis, fuzzy differential subordination results are obtained using a new integral operator introduced in this paper using the well-known confluent hypergeometric function, also known as the Kummer hypergeometric function. The new hypergeometric integral operator is defined by choosing particular parameters, having as inspiration the operator studied by Miller, Mocanu and Reade in 1978. Theorems are stated and proved, which give corollary conditions such that the newly-defined integral operator is starlike, convex and close-to-convex, respectively. The example given at the end of the paper proves the applicability of the obtained results.

Energy-minimal Principles in Geometric Function Theory.

We survey a number of recent developments in geometric analysis as they pertain to the calculus of variations and extremal problems in geometric function theory following the NZMRI lectures given by the first author at those workshops in Napier in 1998 and 2005.

On a new linear operator formulated by Airy functions in the open unit disk

In this note, we formulate a new linear operator given by Airy functions of the first type in a complex domain. We aim to study the operator in view of geometric function theory based on the subordination and superordination concepts. The new operator is suggested to define a class of normalized functions (the class of univalent functions) calling the Airy difference formula. As a result, the suggested difference formula joining the linear operator is modified to different classes of analytic functions in the open unit disk.

Coefficient bounds in the class of functions associated with $q$-function theory

In this paper, we use the concept of $q$-calculus in geometric function theory. For some $\alpha$, $\alpha\in [0,1)$, we consider normalized analytic functions $f$ such that $f’(z)/{\rm d}_qf(z)$ lies in half-plane $\{w:\mathfrak {Re}\ w>\alpha\}$ for all $z$, $|z|< 1$. Here ${\rm d}_q$ is the Jackson $q$-derivative operator well-known in the $q$-calculus theory. The paper is devoted to the coefficient problems of such functions for real and for complex numbers $q$. Coefficient bounds are of particular interest, because of them some geometrical properties of the function can be obtained.

Teichmuller space theory and classical problems of geometric function theory

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions.