Univalent functions. A primer
The authors mainly concentrate on different subclasses of univalent functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\) in the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) and primarily address the book to researchers who are new to the field of geometric function theory. They describe the ideas and techniques used in solving extremal problems and update a variety of results on classes of conformal mappings. Some of complicated proofs are omitted. The contents include 16 chapters. The first chapter gives an elementary introduction to the area of univalent functions, the Bieberbach conjecture and growth and distortion theorems. Star-like, convex, close-to-convex and other functions are defined in Chapter 2. All these function sets relate to analytic functions \(p(z)=1+\sum_{n=1}^{\infty}p_nz^n\) with positive real part in \(\mathbb D\) that are described in Chapter 3. Also, the concept of subordination is presented in Chapter 3 together with Littlewood's subordination theorem, Rogosinski's domination results, theorems by Mocanu and Nunokawa and the useful Clunie-Jack lemma. The two largest Chapters 4 and 9 are devoted to the basic classes of star-like and convex functions and close-to-convex functions, respectively. The coefficient theorems of Chapter 4 give estimates for \(|a_n|\), \(|a_3|-|a_2|\), \(||a_{n+1}|-|a_n||\), \(|a_3-\mu a_2^2|\) with \(\mu\in\mathbb C\), and the Hankel second-order and third-order determinants in the class \(S^*\) of star-like functions. Refined growth theorems give estimates for \(\text{Re}\{zf'(z)/f(z)\}\) and \(|zf'(z)/f(z)|\) dependent on \(|f(z)|\). The same is done for the class \(C\) of convex functions. In addition, some subclasses of \(S^*\) are studied in a similar manner. An analytic function \(f\) in \(\mathbb D\) is said to be star-like of order \(\alpha\) or convex of order \(\alpha\), \(0\leq\alpha<1\), if \[ \text{Re}\frac{zf'(z)}{f(z)}>\alpha\quad \text{or}\quad\text{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha,\;\;\;z\in\mathbb D, \] respectively. The case \(\alpha=0\) corresponds to the definitions of star-like and convex functions. An analytic function \(f\) in \(\mathbb D\) is said to be strongly star-like of order \(\beta\) or convex of order \(\beta\), \(0<\beta\leq1\), if \[ \left|\arg\frac{zf'(z)}{f(z)}\right|<\frac{\pi\beta}{2}\quad\text{or}\quad\left|\arg\left(1+\frac{zf''(z)}{f'(z)}\right)\right|<\frac{\pi\beta}{2}, \;\;\;z\in\mathbb D, \] respectively. An analytic function \(f\) in \(\mathbb D\) is said to be \(\alpha\)-convex, \(\alpha\in\mathbb R\), if \[ \text{Re}\left\{(1-\alpha)\frac{zf'(z)}{f(z)}+\alpha\left(1+\frac{zf''(z)}{f'(z)}\right)\right\}>0,\;\;\;z\in\mathbb D. \] A multiplicative combination of \(\gamma\) and \((1-\gamma)\) powers of \(zf'(z)/f(z)\) and \(1+zf''(z)/f'(z)\) leads to the analogous classes of the so-called \(\gamma\)-star-like functions. In Chapters 5--8, the standard growth and distortion theorems and coefficient estimates are proved for these classes. We say that an analytic function \(f\) in \(\mathbb D\) belongs to the subclass \(B(\alpha)\), \(\alpha\geq0\), of Bazilevich functions if there exists \(g\in S^*\) such that \[ \text{Re}\left[\frac{zf'(z)}{f^{1-\alpha}(z)g^{\alpha}(z)}\right]>0,\;\;\;z\in\mathbb D. \] The class \(B(1)\) coincides with the subclass of close-to-convex functions defined in Chapter 9. We say that a function \(f\) in \(B(\alpha)\) belongs to \(B_1(\alpha)\) if \(g(z)\equiv z\). The classes \(B(1)\), \(B(\alpha)\) and \(B_1(\alpha)\) are studied carefully in Chapters 9--11 following the scheme elaborated earlier. A subclass \(U(\lambda)\), \(0<\lambda\leq1\), consisting of functions \(f\) defined by \(|z^2f'(z)/f^2(z)-1|<\lambda\) in \(\mathbb D\) is discussed in Chapter 12. Chapter 13 deals with the idea of convolution \(f*g\) of functions \(f(z)=\sum_{n=1}^{\infty}a_nz^n\) and \(g(z)=\sum_{n=1}^{\infty}b_nz^n\), \(z\in\mathbb D\), given by \((f*g)(z)=\sum_{n=1}^{\infty}a_nb_nz^n\). Pólya and Schoenberg conjectured that if \(f,g\in C\), then \(f*g\in C\). The authors present the results of Ruscheweyh and Sheil-Small that prove, in particular, the Pólya-Schoenberg conjecture and clarify the connection between the convolution concept and the subordination concept. Meromorphic univalent functions appear in Chapter 14 together with meromorphic star-like, close-to-convex and Bazilevich functions. In order to introduce the reader to one of the most significant advanced methods, the authors outline the main statements after the Loewner theory in Chapter 15. Other topics concerning harmonic univalent functions, bi-univalent functions, the bounded boundary rotation property, differential subordination and differential and integral operators are briefly touched in the final Chapter 16.
scientific article; zbMATH DE number 6840584