Earthline Journal of Mathematical Sciences

On Geometry of Some Subspaces in Hornich Space

Xiaobin Wu — 2024-12-13

Let $\calH$ be the collection of all locally univalent analytic functions $f$ defined on the unit disk $\mathbb{D}$ with the normalization $f(0)=f^{\prime}(0)-1=0$, and $\calS\subseteq \mathcal{H}$ be the class of all univalent functions. For $f,g\in \mathcal{H}$ and $r\in \mathbb{C}$, the Hornich operators are defined as

$$r\odot f(z):=\int_0^z\{f'(\xi)\}^r\mathrm{d}\xi \quad\text{and}\quad f\oplus g(z):=\int_0^zf'(\xi)g'(\xi)\mathrm{d}\xi.$$

We study geometric properties of some subclasses of $\calS$ in the sense of the Hornich space $(\mathcal{H},\odot , \oplus )$. In fact, we prove that the classes of strongly convex functions of order $\beta$, Noshiro-Warschawski functions, and strongly Ozaki close-to-convex functions are all convex in $(\mathcal{H},\odot , \oplus )$, which generalize some known results. Meanwhile, for $M,N\in\calS$, let $T[M,N]:=\{(r,s)\in\mathbb{C}^2:r \odot f \oplus s \odot g\in N,\ for\ \forall f,g\in M \}$. We give the precise descriptions of $T[M,N]$ for some $M,N\in\calS$.

Optimal Debt Ratio and Investment-Consumption Strategies with Taxation in the Presence of Jump Risks

C. B. Ibe — 2024-12-16

This paper derives an optimal debt ratio, consumption rate and investment strategies with taxation for an investor who invest under four background risks: investment, taxation, income and jump risks. The underlying assets considered in this paper are a riskless and risky asset. The risky asset is assumed to follow a jump-diffusion process. We also assume that the income growth rate and tax payment of the investor follow a jump-diffusion process. The aim of the investor is to derive the wealth-after-tax process. The wealth-after-tax process of the investor is taken to be the difference between the wealth-before-tax and the tax payment processes of the investor. The resulting wealth-after-tax process of the investor was solved using dynamic programming approach. As a result, we derive the optimal investment strategies, optimal debt ratio and optimal consumption rate for the investor over time by assuming that the investor chooses a power utility function. The optimal investment strategies were found to involve four components: a speculative portfolio, a tax risks hedging portfolio strategy, an income growth rate risks hedging portfolio strategy and a risk-free fund that holds only the riskless asset. Interestingly, we found that before loan is taken or given, the following must be considered: interest rate on loan to be taken or given, the nominal interest rate, income growth rate, coefficient of the investor willingness to bear the risk of taking debt. We also found that as the income growth rate of the investor increases, the debt profile of the investor decreases. We observe that as the coefficient of risk aversion with respect to debt ratio tends to unity, the amount of debt will be unbearable. It was also observed that the higher an investor willingness to bear the risk of taking debt, the smaller the optimal debt ratio of the investor over time. We further found that when tax rate increases, consumption rate decreases and vice versa. To ascertain the validity of our models, data were collected from six companies in Nigerian Stocks Exchange, and SPSS package was used to analyze the data. Some empirical results were obtained in this paper, using MATLAB software.

Differential Sandwich Theorems for Mittag-Leffler Function Associated with Fractional Integral Defined by Convolution Structure

Noor Yasser Jabir — 2024-12-18

In this work, we use fractional integral and Mittag-Leffler function to obtain some results related to differential subordination and superordination defined by Hadamard product for univalent analytic functions defined in the open unit disk. These results are applied to obtain differential sandwich results. Our results extend corresponding previously known results.