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Register a new userHarmonic spirallike functions and harmonic strongly starlike functions
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Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $lambda$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $lambda$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.
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Let $h^infty_v(mathbf D)$ and $h^infty_v(mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional case let $u (re^{ita},xi) = sum_{j=0}^infty (a_{j0} xi_{j0} r^j cos jtheta +a_{j1} xi_{j1} r^j sin jtheta)$ where $xi ={xi_{ji}}%_{k=0}^infty $ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji} $ which imply that $u$ is in $h^infty_v(mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu.
Kalantaris Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + sum_{j=k}^n a_jleft(z-z_0right)^j,;a_0a_ka_n eq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $frac{pi}{k}$, alternating between arguments of ascent (angles $theta$ where $|p(z_0 + te^{itheta})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantaris original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.
Let $es$ be the family of analytic and univalent functions $f$ in the unit disk $D$ with the normalization $f(0)=f(0)-1=0$, and let $gamma_n(f)=gamma_n$ denote the logarithmic coefficients of $fin {es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $F(c)$ and $G(delta)$ of functions $fin {es}$ defined by $$ {rm Re} left ( 1+frac{zf(z)}{f(z)}right )>1-frac{c}{2}, mbox{ and } , {rm Re} left ( 1+frac{zf(z)}{f(z)}right )<1+frac{delta}{2},quad zin D $$ for some $cin(0,3]$ and $deltain (0,1]$, respectively. We obtain the sharp upper bound for $|gamma_n|$ when $n=1,2,3$ and $f$ belongs to the classes $F(c)$ and $G(delta)$, respectively. The paper concludes with the following two conjectures: begin{itemize} item If $finF (-1/2)$, then $ displaystyle |gamma_n|le frac{1}{n}left(1-frac{1}{2^{n+1}}right)$ for $nge 1$, and $$ sum_{n=1}^{infty}|gamma_{n}|^{2} leq frac{pi^2}{6}+frac{1}{4} ~{rm Li,}_{2}left(frac{1}{4}right) -{rm Li,}_{2}left(frac{1}{2}right), $$ where ${rm Li}_2(x)$ denotes the dilogarithm function. item If $fin G(delta)$, then $ displaystyle |gamma_n|,leq ,frac{delta}{2n(n+1)}$ for $nge 1$. end{itemize}
In this paper we determine the region of variability for spirallike funcions with respect to a boundary point. In the final section we graphically illustrate the region of variability for several sets of parameters.
We prove an analogue of Yaus Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or non-negative subharmonic functions of class Lq, 1<=q<infty, on any graph, and a quantitative version for q > 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.
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