Computational Methods and Function Theory

Li, Ming; Sugawa, Toshiyuki

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0177-8

In this note, we investigate the supremum and the infimum of the functional $$|a_{n+1}|-|a_{n}|$$ | a n + 1 | - | a n | for functions, convex and analytic on the unit disk, of the form $$f(z)=z+a_2z^2+a_3z^3+\cdots .$$ f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ . We also consider the related problem of maximizing the functional $$|a_{n+1}-a_{n}|$$ | a n + 1 - a n | for convex functions f with $$f''(0)=p$$ f ′ ′ ( 0 ) = p for a prescribed $$p\in [0,2].$$ p ∈ [ 0 , 2 ] .

Gundersen, Gary

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0178-7

Twenty-eight research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.

Ariza García, Eusebio; Teodoro, Antonio; Sapiain, María; Vargas, Franklin

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0182-y

Consider the initial value problem $$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$ ∂ t u = L ( t , x , u , ∂ x i u ) , u ( 0 , x ) = φ ( x ) , where t is the time, $${\mathcal L}$$ L is a linear first-order differential operator and $$\varphi $$ φ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator $${\mathcal L}$$ L under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and $$\lambda $$ λ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases $${\mathcal A}^{*}_{2,2}$$ A 2 , 2 ∗ and $${\mathcal A}^{*}_{3,2}$$ A 3 , 2 ∗ . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator $${\mathcal L}$$ L and for any generalized q-metamonogenic initial function $$\varphi $$ φ , provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).

Uğurlu, Ekin; Bairamov, Elgiz

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0185-8

In this paper, we introduce a new approach for the spectral analysis of a linear second order dissipative differential operator with distributional potentials. This approach is related with the inverse operator. We show that the inverse operator is a non-selfadjoint trace class operator. Using Lidskiĭ’s theorem, we introduce a complete spectral analysis of the second order dissipative differential operator. Moreover, we give a trace formula for the trace class integral operator.

Technau, Marc; Technau, Niclas

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0179-6

It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union $$\Gamma $$ Γ of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of $$\Gamma $$ Γ , our approach allows us to derive a Loewner equation for infinitely many slits as well.

Arango, Juan; Arbeláez, Hugo; Mejía, Diego

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0181-z

We present the notion of lower spherical order for locally injective meromorphic functions in the unit disk, and study some properties of functions with positive lower spherical order.

Liu, ZhiHong; Ponnusamy, Saminathan

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0180-0

We consider the convolution of half-plane harmonic mappings with respective dilatations $$(z+a)/(1+az)$$ ( z + a ) / ( 1 + a z ) and $$e^{i\theta }z^{n}$$ e i θ z n , where $$-1<a<1$$ - 1 < a < 1 and $$\theta \in \mathbb {R},n\in \mathbb {N}$$ θ ∈ R , n ∈ N . We prove that such convolutions are locally univalent for $$n=1$$ n = 1 , which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $$n\ge 2$$ n ≥ 2 .

Hu, Qinghua; Li, Songxiao; Shi, Yecheng

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0184-9

In this paper, we give a new characterization for the boundedness, compactness and essential norm of differences of weighted composition operators between weighted-type spaces.

Crowdy, Darren

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0186-7

A covering map formalism for studying the spectral curves associated with finite gap Jacobi matrices is presented. We advocate a constructive function theoretic framework based on use of the Schottky–Klein prime function. The single gap, or genus-one, case is studied in explicit detail.

Karp, D.; Prilepkina, E.

2016 Computational Methods and Function Theory

doi: 10.1007/s40315-016-0183-x

In this paper, we find several new properties of a class of Fox’s H functions which we call delta neutral. In particular, we find an expansion in the neighborhood of the finite non-zero singularity and give new Mellin transform formulas under a special restriction on parameters. The last result is applied to prove a conjecture regarding the representing measure for gamma ratio in Bernstein’s theorem. Furthermore, we find the weak limit of measures expressed in terms of the H function which furnishes a regularization method for integrals containing the delta neutral and zero-balanced cases of Fox’s H function. We apply this result to extend a recently discovered integral equation to the zero-balanced case. In the last section of the paper, we consider a reduced form of this integral equation for Meijer’s G function. This leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function.