Abstract
Function $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$, normalized, analytic and univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class $\mathcal{U}$. if, and only if, \[ \left| \left(\frac{z}{f(z)}\right)^2 -1\right|<1 \quad\quad (z\in \mathbb D). \] In this paper, we prove the Zalcman and the generalized Zalcman conjecture for the class $\mathcal{U}$ and some values of parameters in the conjectures.
Results & Lemmas (4)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1 · coeff
Lemma 1. ([4]). For each function f in, there exists function, analytic in the unit disk, such that and for all, with (2) Additionally,…
Lemma 1. ([4]). For each function f in $\mathcal{U}$ , there exists function $\omega_1$ , analytic in the unit disk, such that $|\omega_1(z)| \leq |z| < 1$ and $|\omega_1'(z)| \leq 1$ for all $z \in \mathbb{D}$ , with
(2)
$$\frac{z}{f(z)} = 1 - a_2 z - z\omega_1(z).$$
Additionally, for $\omega_1(z) = c_1 z + c_2 z^2 + \cdots$ ,
$$|c_1| \le 1$$
, $|c_2| \le \frac{1}{2}(1 - |c_1|^2)$ and $|c_3| \le \frac{1}{3}\left(1 - |c_1|^2 - \frac{4|c_2|^2}{1 + |c_1|}\right)$ .
Let note that for functions f from $\mathcal{U}$ , of form (1), from Lemma 1 we have
$$z = [1 - a_2 z - z\omega_1(z)] \cdot f(z),$$
and after equating the coefficients,
$$a_3 = c_1 + a_2^2,$$
$a_4 = c_2 + 2a_2c_1 + a_2^3,$
$a_5 = c_3 + 2a_2c_2 + c_1^2 + 3a_2^2c_1 + a_2^4.$
Theorem 1 · coeff
Theorem 1. Let be of the form (1). Then These inequalities are sharp with equality for the Koebe function and its rotations. Proof. (i)…
Theorem 1. Let $f \in \mathcal{U}$ be of the form (1). Then
$$\begin{array}{ll} (i) & |a_2^2 - a_3| \leq 1; \\ (ii) & |a_3^2 - a_5| \leq 4. \end{array}$$
$$(ii) |a_3^2 - a_5| \leq 4$$
These inequalities are sharp with equality for the Koebe function $k(z) = \frac{z}{(1-z)^2}$ $z + \sum_{n=2} nz^n$ and its rotations.
Proof.
(i) From
$$a_3 = c_1 + a_2^2$$
we have $|a_2^2 - a_3| = |-c_1| \le 1$ .
(ii) From the Bieberbach conjecture, $|a_3| = |c_1 + a_2^2| \le 3$ , and further calculations show that
$$\begin{aligned} \left| a_3^2 - a_5 \right| &= \left| c_3 + 2a_2c_2 + a_2^2c_1 \right| \\ &= \left| c_3 + 2a_2c_2 - c_1^2 + c_1(c_1 + a_2^2) \right| \\ &\leq \left| c_3 \right| + 2\left| a_2 \right| \left| c_2 \right| + \left| c_1 \right|^2 + \left| c_1 \right| \left| c_1 + a_2^2 \right| \\ &\leq \left| c_3 \right| + 2\left| a_2 \right| \left| c_2 \right| + \left| c_1 \right|^2 + 3\left| c_1 \right| \\ &\leq \frac{1}{3} \left( 1 - \left| c_1 \right|^2 - \frac{4\left| c_2 \right|^2}{1 + \left| c_1 \right|} \right) + 4\left| c_2 \right| + \left| c_1 \right|^2 + 3\left| c_1 \right| \\ &:= f_1(\left| c_1 \right|, \left| c_2 \right|), \end{aligned}$$
where
$$f_1(x,y) = \frac{1}{3} \left( 1 - x^2 - \frac{4y^2}{1+x} \right) + 4y + x^2 + 3x,$$
$0 \le x = |c_1| \le 1$ and $0 \le y = |c_2| \le \frac{1}{2}(1 - x^2)$ , i.e., $(x, y) \in G := [0, 1] \times [0, (1 - x^2)/2]$ .
Since, $\frac{\partial f_1}{\partial y}(x,y) = \frac{4}{3} \left(\frac{y}{1+x}\right)^2 + \frac{4}{3}x + 3 > 0$ for all $(x,y) \in G$ , we have that there are no singular points in the interior of G and $f_1$ attains its maximum on the boundary of G.
Further, for x = 0 we have $0 \le y \le \frac{1}{2}$ and $f_1(0, y) = \frac{1}{3}(1 - 4y^2) + 4y \le 2$ .
Also, for $0 \le x \le 1$ and y = 0 we have $f_1(x, 0) = \frac{1}{3}(1 - x^2) + x^2 + 3x \le 4$ .
Finally, for $0 \le x \le 1$ and $y = \frac{1}{2}(1-x^2)$ we have $f_1(x, \frac{1}{2}(1-x^2)) = 2 + \frac{10}{3}x - x^2 - \frac{1}{3}x^3 \le 4$ , since the last function is an increasing one on [0, 1].
3. Generalized Zalcman conjecture for the class $\mathcal U$
In this section we give direct proof of the generalized Zalcman conjecture for the class $\mathcal{U}$ for the cases m=2, n=3; and m=2, n=4.
Theorem 2 · coeff
Theorem 2. Let be of the form (1). Then - (i); - These inequalities are sharp with equality for the Koebe function and its rotations.…
Theorem 2. Let $f \in \mathcal{U}$ be of the form (1). Then
- (i) $|a_2a_3-a_4| \leq 2$ ;
- $(ii) |a_2a_4-a_5| \leq 3.$
These inequalities are sharp with equality for the Koebe function $k(z) = \frac{z}{(1-z)^2} = z + \sum_{n=2} nz^n$ and its rotations.
Proof.
(i) In this case we have
$$|a_2a_3 - a_4| = |c_2 + a_2c_1| \le |c_2| + |a_2||c_1| \le |c_2| + 2|c_1|$$
$$\le \frac{1}{2}(1 - |c_1|^2) + 2|c_1| \le \frac{1}{2}(1 - |c_1|^2 + 4|c_1|) \le 2.$$
(ii) In a similar way as in the proof of Theorem 1(ii),
$$\begin{aligned} |a_4 a_2 - a_5| &= \left| c_3 + a_2 c_2 + a_2^2 c_1 + c_1^2 \right| \\ &\leq |c_3| + |a_2| |c_2| + |c_1| |a_2^2 + c_1| \\ &\leq |c_3| + |a_2| |c_2| + 3 |c_1| \\ &\leq \frac{1}{3} \left( 1 - |c_1|^2 - \frac{4|c_2|^2}{1 + |c_1|} \right) + 2|c_2| + 3|c_1| \\ &:= f_2(|c_1|, |c_2|), \end{aligned}$$
where
$$f_2(x,y) = \frac{1}{3} \left( 1 - x^2 - \frac{4y^2}{1+x} \right) + 2y + 3x,$$
$0 \le x = |c_1| \le 1$ and $0 \le y = |c_2| \le \frac{1}{2}(1 - x^2)$ , i.e., $(x, y) \in G :=$ $[0,1] \times [0,(1-x^2)/2].$
Again, $\frac{\partial f_2}{\partial y}(x,y) = \frac{4}{3} \left(\frac{y}{1+x}\right)^2 - \frac{2}{3}x + 3 > 0$ for all $(x,y) \in G$ , so $f_2$ attains its maximum on the boundary of G.
The conclusion follows since on the edges of G we have:
- x = 0, $0 \le y \le \frac{1}{2}$ and $f_2(0, y) = \frac{1}{3}(1 4y^2) + 2y \le 1$ ; y = 0, $0 \le x \le 1$ and $f_2(x, 0) = \frac{1}{3}(1 x^2) + 3x \le 3$ ; $y = \frac{1}{2}(1 x^2)$ , $0 \le x \le 1$ and $f_2(x, \frac{1}{2}(1 x^2)) = 1 + \frac{10}{3}x x^2 \frac{1}{3}x^3 \le 3$ .
Theorem 3 · coeff
Theorem 3. Let be of the form (1). Then - (i); - (ii). These inequalities are sharp with equality for the Koebe function and its rotations.…
Theorem 3. Let $f \in \mathcal{U}$ be of the form (1). Then
- (i) $|a_4 a_2^3| \le 4$ ;
- (ii) $|a_5 a_2^4| \le 11$ .
These inequalities are sharp with equality for the Koebe function $k(z) = \frac{z}{(1-z)^2} =$ $z + \sum_{n=2} nz^n$ and its rotations.
Proof.
(i) It is easy to verify that
$$|a_4 - a_2^3| = |c_2 + 2a_2c_1| \le |c_2| + 2|a_2||c_1|$$
$$\le \frac{1}{2}(1 - |c_1|^2) + 4|c_1| = \frac{1}{2}(1 + 8|c_1| - |c_1|^2) \le 4.$$
(ii) We will again use that $|a_3| = |c_1 + a_2| < 3$ and receive
$$\begin{aligned} |a_5 - a_2^4| &= |c_3 + 2a_2c_2 + c_1^2 + 3a_2^2c_1| \\ &= |c_3 + 2a_2c_2 - 2c_1^2 + 3c_1(c_1 + a_2^2)| \\ &\leq |c_3| + 2|a_2||c_2| + 2|c_1|^2 + 9|c_1| \\ &\leq \frac{1}{3} \left( 1 - |c_1|^2 - \frac{4|c_2|^2}{1 + |c_1|} \right) + 4|c_2| + 2|c_1|^2 + 9|c_1| \\ &:= g(|c_1|, |c_2|), \end{aligned}$$
where
$$g(x,y) = \frac{1}{3} \left( 1 - x^2 - \frac{4y^2}{1+x} \right) + 4y + 2x^2 + 9x,$$
$$0 \le x = |c_1| \le 1$$
and $0 \le y = |c_2| \le \frac{1}{2}(1 - x^2)$ , i.e., $(x, y) \in G := [0, 1] \times [0, (1 - x^2)/2]$ .
Since, $\frac{\partial g}{\partial y}(x,y) = \frac{10}{3}x + \frac{4}{3}\left(\frac{y}{1+x}\right)^2 + 9 > 0$ for all $(x,y) \in G$ , so g has no critical points in the interior of G and attains its maximum on the
- x = 0, $0 \le y \le \frac{1}{2}$ and $g(0, y) = \frac{1}{3}(1 + 12y 4y^2) \le 2$ ; y = 0, $0 \le x \le 1$ and $g(x, 0) = \frac{5}{3}x^2 + 9x + \frac{1}{3} \le 11$ ; $y = \frac{1}{2}(1 x^2)$ , $0 \le x \le 1$ and $g(x, \frac{1}{2}(1 x^2)) = 2 + \frac{28}{3}x \frac{1}{3}x^3 \le 11$ .
The statement (ii) follows directly.
References
- <span id="page-4-1"></span><span id="page-4-0"></span>1. L. De Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1–2, 137–152.
- 2. S.L. Krushkal, A short geometric proof of the Zalcman and Bieberbach conjectures, preprint, arXiv:1408.1948.
- <span id="page-4-2"></span>3. W. Ma, Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl., 234 (1999), pp. 328-339
- <span id="page-4-5"></span>4. M. Obradović, N. Tuneski, Some properties of the class U, Annales. Universitatis Mariae Curie-Skodowska. Sectio A - Mathematica, Vol. 73 No.1, (2019), 49-56.
- <span id="page-4-3"></span>5. V. Ravichandran, S. Verma, Generalized Zalcman conjecture for some classes of analytic functions. J. Math. Anal. Appl. 450 (2017), no. 1, 592-605.
- <span id="page-4-4"></span>6. D.K. Thomas, N. Tuneski, A. Vasudevarao, Univalent Functions: A Primer, De Gruyter Studies in Mathematics 69, De Gruyter, Berlin, Boston, 2018.
DEPARTMENT OF MATHEMATICS, FACULTY OF CIVIL ENGINEERING, UNIVERSITY OF BELGRADE. Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia
E-mail address: obrad@grf.bg.ac.rs
DEPARTMENT OF MATHEMATICS AND INFORMATICS, FACULTY OF MECHANICAL ENGINEERING, Ss. Cyril and Methodius University in Skopje, Karpoš II b.b., 1000 Skopje, Republic of NORTH MACEDONIA.
E-mail address: nikola.tuneski@mf.edu.mk
Coefficient bounds & claims (7)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|a_2^2 - a_3| (Zalcman n=2) ≤ 1 for class U (sharp) [Theorem 1 (i)]
coefficient_bound
|a_3^2 - a_5| (Zalcman n=3) ≤ 4 for class U (sharp) [Theorem 1 (ii)]
coefficient_bound
|a_2*a_3 - a_4| (generalized Zalcman m=2,n=3) ≤ 2 for class U (sharp) [Theorem 2 (i)]
coefficient_bound
|a_2*a_4 - a_5| (generalized Zalcman m=2,n=4) ≤ 3 for class U (sharp) [Theorem 2 (ii)]
coefficient_bound
|a_4 - a_2^3| (Krushkal n=4,p=1) ≤ 4 for class U (sharp) [Theorem 3 (i)]
coefficient_bound
|a_5 - a_2^4| (Krushkal n=5,p=1) ≤ 11 for class U (sharp) [Theorem 3 (ii)]
function_family
Class U: f in A : |(z/f(z))^2 * f'(z) - 1| < 1 for z in D; functions are univalent but not starlike in general
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