By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

On the theory of multivalent functions

For analytic functions f in the unit disk $${\mathbb {D}}$$ normalized by $$f(0)=0$$ and $$f'(0)=1$$ satisfying in $${\mathbb {D}}$$ respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$ the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$ is real was computed.

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On the third logarithmic coefficient in some subclasses of close-to-convex functions

\n We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

Second hankel determinant of logarithmic coefficients of convex and starlike functions

Coefficient Estimates for Certain Classes of Analytic Functions (Inequalities in Univalent Function Theory and Its Applications)

Let denote the class of analytic and univalent functions in which are of the form . We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in and certain of its subclasses. We also discuss similar problems for typically real functions.

Toeplitz determinants whose elements are the coefficients of analytic and univalent functions

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} \gamma_n z^n$. Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, {\it Proc. Amer. Math. Soc.} {\bf 144} (2016), 1681--1687] proved that $|\gamma_3|\le \frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and claimed that the estimate is sharp by providing a form of a extremal function. In the present paper, we pointed out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument $0$). We also determine a sharp upper bound of $|\gamma_3|$ for close-to-convex functions (with argument $0$) with respect to the Koebe function.

On logarithmic coefficients of some close-to-convex functions

The logarithmic coefficients of an analytic and univalent function in the unit disc with the normalisation are defined by . In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of , , for such functions .

Logarithmic coefficients of some close-to-convex functions

For any real number ${\\it\\beta}$ with ${\\it\\beta}>1$, let ${\\mathcal{M}}(\\,{\\it\\beta})$ (${\\mathcal{N}}(\\,{\\it\\beta})$ respectively) denote the class of analytic functions $f$ in the unit disk $\\mathbb{D}:=\\{z\\in \\mathbb{C}:|z|<1\\}$ of the form $f(z)=z+\\sum _{n=2}^{\\infty }a_{n}z^{n}$ and satisfying $\\text{Re}\\,P_{f}<{\\it\\beta}$ ($\\text{Re}\\,Q_{f}<{\\it\\beta}$ respectively) in $\\mathbb{D}$, where $P_{f}=zf^{\\prime }(z)/f(z)$ and $Q_{f}=1+zf^{\\prime \\prime }(z)/f^{\\prime }(z)$. Also, for ${\\it\\beta}>1$, let ${\\mathcal{M}}{\\rm\\Sigma}(\\,{\\it\\beta})$ (${\\mathcal{N}}{\\rm\\Sigma}(\\,{\\it\\beta})$ respectively) denote the class of analytic functions $g$ of the form $g(z)=z(1+\\sum _{n=1}^{\\infty }b_{n}z^{-n})$ and satisfying $\\text{Re}\\,P_{g}<{\\it\\beta}$ ($\\text{Re}\\,Q_{g}<{\\it\\beta}$ respectively) for $z\\in {\\rm\\Delta}=\\{z\\in \\mathbb{C}:1<|z|<\\infty \\}$. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional ${\\rm\\Lambda}_{{\\it\\lambda}}(f)=a_{3}-{\\it\\lambda}a_{2}^{2}$ for functions $f$ in the classes ${\\mathcal{M}}(\\,{\\it\\beta})$ and ${\\mathcal{N}}(\\,{\\it\\beta})$. Further, we shall solve the maximal area problem for functions of the type $z/f(z)$ when $f\\in {\\mathcal{M}}(\\,{\\it\\beta})$, which is Yamashita’s conjecture for the class ${\\mathcal{M}}(\\,{\\it\\beta})$. We shall obtain the radius of convexity for the class ${\\mathcal{N}}(\\,{\\it\\beta})$. We shall also determine the coefficient bounds for functions $g$ in the classes ${\\mathcal{M}}{\\rm\\Sigma}(\\,{\\it\\beta})$ and ${\\mathcal{N}}{\\rm\\Sigma}(\\,{\\it\\beta})$ and the inverse coefficient bounds for functions $g$ in the class ${\\mathcal{M}}{\\rm\\Sigma}(\\,{\\it\\beta})$. All the results are sharp.

Coefficient inequalities and yamashita’s conjecture for some classes of analytic functions

For a normalized analytic function -fold symmetric starlike functions of complex order defined by a subordination relation. One of the particular cases of the latter problem includes the solution to a conjecture of Yamashita, which was settled recently by Obradovic et al. [‘A proof of Yamashita’s conjecture on area integral’, Comput. Methods Funct. Theory13 (2013), 479–492].

Firoz Ali

Papers

Toeplitz determinants whose elements are the coefficients of analytic and univalent functions

On logarithmic coefficients of some close-to-convex functions

Logarithmic coefficients of some close-to-convex functions

Coefficient inequalities and yamashita’s conjecture for some classes of analytic functions

Integral means and dirichlet integral for certain classes of analytic functions