REsources
GFT Books
Duren, Peter L. Univalent functions. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 259. Springer-Verlag, New York, 1983. xiv+382 pp. ISBN: 0-387-90795-5 MR0708494
Duren, Peter. Harmonic mappings in the plane. Cambridge Tracts in Mathematics, 156. Cambridge University Press, Cambridge, 2004. xii+212 pp. ISBN: 0-521-64121-7 MR2048384
Goluzin, G. M. Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, Vol. 26 American Mathematical Society, Providence, R.I. 1969 vi+676 pp. MR0247039
Goodman, A. W. Univalent functions. Vol. I. Mariner Publishing Co., Inc., Tampa, FL, 1983. {\rm xvii}+246 pp. ISBN: 0-936166-10-X MR0704183
Goodman, A. W. Univalent functions. Vol. II. Mariner Publishing Co., Inc., Tampa, FL, 1983. {\rm xii}+311 pp. ISBN: 0-936166-11-8 MR0704184
Graham, Ian; Kohr, Gabriela. Geometric function theory in one and higher dimensions. Monographs and Textbooks in Pure and Applied Mathematics, 255. Marcel Dekker, Inc., New York, 2003. xviii+530 pp. ISBN: 0-8247-0976-4 MR2017933
Hallenbeck, D. J.; MacGregor, T. H. Linear problems and convexity techniques in geometric function theory. Monographs and Studies in Mathematics, 22. Pitman (Advanced Publishing Program), Boston, MA, 1984. {\rm xvii}+182 pp. ISBN: 0-273-08637-5 MR0768747
Hayman, W. K. Research problems in function theory. The Athlone Press [University of London], London, 1967 vii+56 pp. MR0217268
Krantz, Steven G. Geometric function theory. Explorations in complex analysis. Cornerstones. Birkhäuser Boston, Inc., Boston, MA, 2006. xiv+314 pp. ISBN: 978-0-8176-4339-3; 0-8176-4339-7 MR2167675
Miller, Sanford S.; Mocanu, Petru T. Differential subordinations. Theory and applications. Monographs and Textbooks in Pure and Applied Mathematics, 225. Marcel Dekker, Inc., New York, 2000. xii+459 pp. ISBN: 0-8247-0029-5 MR1760285
Pommerenke, Christian. Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbucher, Band XXV. Vandenhoeck & Ruprecht, Gottingen, 1975. 376 pp. MR0507768
Rosenblum, Marvin; Rovnyak, James. Topics in Hardy classes and univalent functions. Birkhauser Advanced Texts: Basler Lehrbucher. [Birkhauser Advanced Texts: Basel Textbooks] Birkhauser Verlag, Basel, 1994. {\rm xii}+250 pp. ISBN: 3-7643-5111-X MR1307384
Ruscheweyh, Stephan. Convolutions in geometric function theory. Fundamental Theories of Physics. Seminaire de Mathematiques Superieures [Seminar on Higher Mathematics], 83. Presses de l'Universite de Montreal, Montreal, Que., 1982. 168 pp. ISBN: 2-7606-0600-7 MR0674296
Books on Special Functions
Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664 pp. ISBN: 0-521-62321-9; 0-521-78988-5 MR1688958
Baricz, Arpad. Generalized Bessel functions of the first kind. Lecture Notes in Mathematics, 1994. Springer-Verlag, Berlin, 2010. xiv+206 pp. ISBN: 978-3-642-12229-3 MR2656410
Bell, W. W. Special functions for scientists and engineers. D. Van Nostrand Co., Ltd., London-Princeton, N.J.-Toronto, Ont., 1968. {\rm xiv}+247 pp. MR0302944
Srivastava, H. M.; Manocha, H. L. A treatise on generating functions. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley \& Sons, Inc.], New York, 1984. 569 pp. ISBN: 0-85312-508-2 MR0750112
Temme, Nico M. Special functions. An introduction to the classical functions of mathematical physics. A Wiley-Interscience Publication. John Wiley \& Sons, Inc., New York, 1996. {\rm xiv}+374 pp. ISBN: 0-471-11313-1 MR1376370
Wang, Z. X.; Guo, D. R. Special functions. Translated from the Chinese by Guo and X. J. Xia. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. {\rm xviii}+695 pp. ISBN: 9971-50-659-9 MR1034956
Watson, G. N. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. {\rm vi}+804 pp. MR0010746
Mathematica Codes
Plotting the image of the unit disk under exponential function
f[z_] := Exp[z]
ff[r_, t_] := f[r Exp[I t]];
u[r_, t_] := Re[ff[r, t]];
v[r_, t_] := Im[ff[r, t]];
ParametricPlot[{u[r, t], v[r, t]}, {r, 0, 1}, {t, 0, 2 Pi}, PlotStyle -> Red, BoundaryStyle -> Blue, MeshStyle -> Black, PlotPoints -> 100, Mesh -> 10, AxesOrigin -> {0, 0}]
Plotting the images of the unit circle under 4 functions: Sqrt[1 + z], Exp[z], Sin[z] and 2/(1+Exp[-z])
ParametricPlot[{{Re[Sqrt[1 + Exp[I t]]], Im[Sqrt[1 + Exp[I t]]]}, {Re[Exp[Exp[I t]]], Im[Exp[Exp[I t]]]}, {Re[1 + Sin[Exp[I t]]], Im[1 + Sin[Exp[I t]]]}, {Re[2/(1 + Exp[-Exp[I t]])], Im[2/(1 + Exp[-Exp[I t]])]},}, {t, -Pi, Pi}, PlotStyle -> {Red, Blue, Green, Yellow}]
Illustration of the sharpness of the radius constant in Corollary 3.1 of the research article [Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38 (2015), no. 1, 365--386. MR3394060]
l[z_] := z/(1 - z)
f[z_] := z l'[z]/l[z];
ff[r_, t_] := f[r Exp[I t]];
u[r_, t_] := Re[ff[r, t]];
v[r_, t_] := Im[ff[r, t]];
Show[ParametricPlot[{u[r, t], v[r, t]}, {r,0, (Exp[1] - 1)/(Exp[1])}, {t, 0, 2 Pi}, PlotStyle -> Gray, BoundaryStyle -> Black, MeshStyle -> Black, PlotPoints -> 100, Mesh -> 10, AxesOrigin -> {0, 0}], ParametricPlot[{Re[Exp[Exp[I t]]], Im[Exp[Exp[I t]]]}, {t, 0, 2 Pi}],PlotRange -> Full]
LaTeX Codes
Making Mark-Strohhacker Implications in LaTeX
\documentclass[12pt, a4paper]{amsart}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\renewcommand{\Re}{\textrm{Re}}
\begin{document}
\begin{figure}[h]
\begin{tikzpicture}
\node[right] at (0,4) {$\Re \dfrac{zf''(z)}{f'(z)}+1>0$};
\node[right] at (5,6) {$\Re \dfrac{zf'(z)}{f(z)}>\dfrac{1}{2}$};
\node[right] at (9.5,4) {$\Re \dfrac{f(z)}{z}>\dfrac{1}{2}$};
\node[right] at (5,2) {$\Re \sqrt{f'(z)}>\dfrac{1}{2}$};
\draw[-Implies,double distance=3pt] (3.75,4.75) -- (5,5.5);
\draw[-Implies,double distance=3pt] (3.75,3.25) -- (5,2.5);
\draw[-Implies,double distance=3pt] (8,5.5) -- (9.25,4.75);
\draw[-Implies,double distance=3pt] (8,2.5) -- (9.25,3.25);
\end{tikzpicture}
\caption{Mark-Strohhacker Implication}
\end{figure}
\end{document}
Making graphs in LaTeX
\documentclass[12pt, a4paper]{amsart}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xmin=0, xmax=2.5*pi,
ymin=-2, ymax=2,
axis lines=middle,
xtick={0, pi/2, pi, 3*pi/2, 2*pi},
xticklabels={$0$, $\pi/2$, $\pi$, $3\pi/2$, $2\pi$}, samples=1000, domain=0: 2*pi,
xmajorgrids=true,
grid style=dashed,
ticklabel style={font=\tiny}]
\addplot[red]{sin(deg(x))}
node[right, pos=0.3]{$\sin x$};
\addlegendentry{$\sin x$}
\addplot[blue]{cos(deg(x))}
node[right, pos=0.9]{$\cos x$};
\addlegendentry{$\cos x$}
\end{axis}
\end{tikzpicture}
\end{document}
Making disks in LaTeX
\documentclass[12pt, a4paper]{amsart}
\usepackage{tikz}
\renewcommand{\Re}{\textrm{Re}}
\renewcommand{\Im}{\textrm{Im}}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\pgfplotsset{width=10cm}
\begin{axis}[font=\tiny,
xmin=-5, xmax=5,
ymin=-5, ymax=5,
xtick={-4,-3,-2,-1,0,1,2,3,4},
ytick={-4,-3,-2,-1,0,1,2,3,4},
xticklabels={$-4$,$-3$,$-2$,$-1$, $0$,$1$,$2$,$3$,$4$},
yticklabels={$-4i$,$-3i$,$-2i$,$-1i$, $0i$,$1i$,$2i$,$3i$,$4i$},
axis equal,
axis lines=middle,
xlabel=$\Re(z)$,
ylabel=$\Im(z)$,
disabledatascaling]
\fill [opacity=0.3] (1,-1) circle [radius=3];
\node [below right] at (1,-1) {$|z-1+i| \leq 3$};
\end{axis}
\end{tikzpicture}
\end{document}
Useful Links
Special Issues on GFT
Advances in Geometric Function Theory with Analytic Function Spaces, Journal of Function Spaces (2022) Hindawi
Complex Analysis and Geometric Function Theory, Mathematics (2022-23) MDPI
Symmetry in Geometric Function Theory, Symmetry (2022-23) MDPI
New Developments in Geometric Function Theory, Axioms (2022-23) MDPI
Developments in Geometric Function Theory, Journal of Function Spaces (2021) Hindawi
Quantum Calculus and Its Applications in Geometric Function Theory, AIMS Mathematics (2021) AIMS Press
Advances in Geometric Function Theory, Journal of Mathematics (2021-22)
Analytic and Harmonic Univalent Functions, Abstract and Applied Analysis (2014) Hindawi
Survey Articles in GFT
Ravichandran, V.; Nagpal, Sumit. A survey on univalent functions with fixed second coefficient. Math. Newsl. 33 (2022), no. 1, 17--28. MR4518875
Zayed, Hanaa M. A survey on fractional calculus in geometric function theory. J. Fract. Calc. Appl. 12 (2021), no. 3, Paper No. 3, 16 pp. MR4294666
Ahuja, Om P.; Çetinkaya, Asena. A survey on the theory of integral and related operators in geometric function theory. Mathematical analysis and computing, 635--652, Springer Proc. Math. Stat., 344, Springer, Singapore, [2021], ©2021. MR4281615
Cho, Nak Eun; Kumar, Virendra; Ravichandran, V. A survey on coefficient estimates for Caratheodory functions. Appl. Math. E-Notes 19 (2019), 370--396. MR3980747
Gregorczyk, Magdalena; Koczan, Leopold. A survey of a selection of methods for determination of Koebe sets. Ann. Univ. Mariae Curie-Skłodowska Sect. A 71 (2017), no. 2, 63--67. MR3742485
Kayumov, I. R. A survey on estimates for the integral means spectrum of conformal mappings. (Russian) Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki 157 (2015), no. 2, 104--115. MR3478789
Sahoo, Pravati; Mohapatra, R. N. A survey on some special classes of Bazilevič functions and related function classes. Current topics in pure and computational complex analysis, 63--88, Trends Math., Birkhäuser/Springer, New Delhi, 2014. MR3329713
Bshouty, Daoud; Lyzzaik, Abdallah. Boundary behavior of univalent harmonic mappings. A survey of recent boundary behavior results of univalent harmonic mappings. Current topics in pure and computational complex analysis, 1--19, Trends Math., Birkhäuser/Springer, New Delhi, 2014. MR3329710
Shareef, Zahid; Hussain, Saqib; Darus, Maslina. Convolution operators in the geometric function theory. J. Inequal. Appl. 2012, 2012:213, 11 pp. MR3016328
Noor, Khalida Inayat; Malik, Bushra; Mustafa, Saima. A survey on functions of bounded boundary and bounded radius rotation. Appl. Math. E-Notes 12 (2012), 136--152. MR2992947
Ravichandran, V. Geometric properties of partial sums of univalent function, Math. Newsl., 22 (2012), no. 3, 208-221.
Ali, Rosihan M.; Ravichandran, V. Uniformly convex and uniformly starlike functions, Math. Newsl., 21 (2011), no. 1, 16-30.
Ahuja, Om P. Planar harmonic univalent and related mappings. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 122, 18 pp. MR2178303
Sahoo, Pravati. A short survey on certain integral operators. J. Orissa Math. Soc. 23/24 (2004/05), 132--145. MR2309643
Kim, Yong Chan. Survey on integral transforms in the univalent function theory. New extension of historical theorems for univalent function theory (Japanese) (Kyoto, 1999). Sūrikaisekikenkyūsho Kōkyūroku No. 1164 (2000), 31--44. MR1805554
Rønning, Frode. A survey on uniformly convex and uniformly starlike functions. Ann. Univ. Mariae Curie-Skłodowska Sect. A 47 (1993), 123--134. MR1344982
Silverman, Herb. A survey with open problems on univalent functions whose coefficients are negative. Rocky Mountain J. Math. 21 (1991), no. 3, 1099--1125. MR1138154
Ahuja, O. P.; Silverman, H. A survey on spiral-like and related function classes. Math. Chronicle 20 (1991), 39--66. MR1137872
Ahuja, O. P. The Bieberbach conjecture and its impact on the developments in geometric function theory. Math. Chronicle 15 (1986), 1--28. MR0900335
MacGregor, T. H. Linear methods in geometric function theory. Amer. Math. Monthly 92 (1985), no. 6, 392--406. MR0795249
Liu, Shu Qin. A survey of developments in some problems concerning univalent functions. (Chinese) Adv. in Math. (Beijing) 11 (1982), no. 2, 115--133. MR0693950
Goodman, A. W. An invitation to the study of univalent and multivalent functions. Internat. J. Math. Math. Sci. 2 (1979), no. 2, 163--186. MR0539196