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SUMMARY:Navneet Lal Sharma (Gati Shakti Vishwavidyalaya)
DTSTART:20230523T130000Z
DTEND:20230523T140000Z
DTSTAMP:20260422T201824Z
UID:CAvid/107
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/107/">
 Estimates logarithmic coefficients for certain classes of univalent functi
 ons</a>\nby Navneet Lal Sharma (Gati Shakti Vishwavidyalaya) as part of CA
 vid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nL
 et $\\mathcal{S}$ be the family of analytic and univalent functions $f$ in
  the unit disk $\\mathbb{D}$\nwith the normalization $f(0)=f'(0)-1=0$.\nTh
 e logarithmic coefficients $\\gamma_n$ of $f\\in \\mathcal{S}$ are defined
  by the formula\n$$\\log\\left(\\frac{f(z)}{z}\\right)\\\,=\\\,2\\sum_{n=1
 }^{\\infty}\\gamma_n(f)z^n.\n$$\nIn this talk\, we will discuss bounds for
  the logarithmic coefficients for certain geometric subfamilies of univale
 nt functions as starlike\, convex\, close-to-convex and Janowski starlike 
 functions. Also\, we consider the families $\\mathcal{F}(c)$ and \n$\\math
 cal{G}(\\delta)$  of functions $f\\in \\mathcal{S}$ defined by\n$$ {\\rm R
 e}  \\left ( 1+\\frac{zf''(z)}{f'(z)}\\right )>1-\\frac{c}{2}\\\, \\mbox{ 
 and } \\\,\n{\\rm Re}  \\left ( 1+\\frac{zf''(z)}{f'(z)}\\right )<1+\\frac
 {\\delta}{2}\,\\quad z\\in \\mathbb{D} $$\nfor some $c\\in(0\,3]$ and $\\d
 elta\\in (0\,1]$\, respectively. We obtain the sharp upper bound for $|\\g
 amma_n|$ when $n=1\,2\,3$ and $f$ belongs to the classes \n$\\mathcal{F}(c
 )$ and $\\mathcal{G}(\\delta)$\, respectively. We conclude with the follow
 ing two conjectures:\n\n* If $f\\in\\mathcal{F}(-1/2)$\, then  $ \\display
 style |\\gamma_n|\\le \\frac{1}{n}\\left(1-\\frac{1}{2^{n+1}}\\right)$\n f
 or $n\\ge 4$\,  and\n$$ \\sum_{n=1}^{\\infty}|\\gamma_{n}|^{2} \\leq \\fra
 c{\\pi^2}{6}+\\frac{1}{4} ~{\\rm Li\\\,}_{2}\\left(\\frac{1}{4}\\right)\n 
   -{\\rm Li\\\,}_{2}\\left(\\frac{1}{2}\\right)\, $$\nwhere ${\\rm Li}_2(x
 )$ denotes the dilogarithm function. \n\n* If $f\\in \\mathcal{G}(\\delta)
 $\, then  $ \\displaystyle |\\gamma_n|\\\,\\leq \\\,\\frac{\\delta}{2n(n+1
 )}$   for $n\\ge 4$.\n\n\nThis talk is based on the following article.\n\n
 S. Ponnusamy\, N. L. Sharma and K.-J. Wirths\,\nLogarithmic coefficients p
 roblems in families related to starlike and convex functions\, . Aust. Mat
 h. Soc.\, 109(2) (2019)\, 230--249.\n
LOCATION:https://researchseminars.org/talk/CAvid/107/
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