Bloch's theorem (complex analysis)

<h2 id="statement">Statement</h2>
Let f be a holomorphic function in the unit disk |z| ≤ 1 for which

|
        
        
          f
          ′
        
        (
        0
        )
        
          |
        
        =
        1
      
    
    {\displaystyle |f'(0)|=1}

Bloch's theorem states that there is a disk S ⊂ D on which f is <a href="/facts/Biholomorphism/3g5kL1RD">biholomorphic</a> and f(S) contains a disk with radius 1/72.

<h2 id="landaus-theorem">Landau's theorem</h2>
If f is a holomorphic function in the unit disk with the property |f′(0)| = 1, then let Lf be the radius of the largest disk contained in the image of f.
Landau's theorem states that there is a constant L defined as the infimum of Lf over all such functions f, and that L is greater than Bloch's constant L ≥ B.
This theorem is named after <a href="/facts/Edmund_Landau/lVgFVLBw">Edmund Landau</a>.

<h2 id="valirons-theorem">Valiron's theorem</h2>
Bloch's theorem was inspired by the following theorem of <a href="/facts/Georges_Valiron/rhGvdJbF">Georges Valiron</a>:
Theorem. If f is a non-constant entire function then there exist disks D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D.
Bloch's theorem corresponds to Valiron's theorem via the so-called <a href="/facts/Bloch%2527s_principle/TyI4ktsP">Bloch's principle</a>.

<h2 id="proof">Proof</h2>
<h3>Landau's theorem</h3>
We first prove the case when f(0) = 0, f′(0) = 1, and |f′(z)| ≤ 2 in the unit disk. 
By <a href="/facts/Cauchy%2527s_integral_formula/4B35nkvV">Cauchy's integral formula</a>, we have a bound

|
        
        
          f
          ″
        
        (
        z
        )
        
          |
        
        =
        
          |
          
            
              
                1
                
                  2
                  π
                  i
                
              
            
            
              ∮
              
                γ
              
            
            
              
                
                  
                    f
                    ′
                  
                  (
                  w
                  )
                
                
                  (
                  w
                  −
                  z
                  
                    )
                    
                      2
                    
                  
                
              
            
            
            
              d
            
            w
          
          |
        
        ≤
        
          
            1
            
              2
              π
            
          
        
        ⋅
        2
        π
        r
        
          sup
          
            w
            =
            γ
            (
            t
            )
          
        
        
          
            
              
                |
              
              
                f
                ′
              
              (
              w
              )
              
                |
              
            
            
              
                |
              
              w
              −
              z
              
                
                  |
                
                
                  2
                
              
            
          
        
        ≤
        
          
            2
            r
          
        
        ,
      
    
    {\displaystyle |f''(z)|=\left|{\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f'(w)}{(w-z)^{2}}}\,\mathrm {d} w\right|\leq {\frac {1}{2\pi }}\cdot 2\pi r\sup _{w=\gamma (t)}{\frac {|f'(w)|}{|w-z|^{2}}}\leq {\frac {2}{r}},}

where γ is the counterclockwise circle of radius r around z, and 0 < r < 1 − |z|.
By <a href="/facts/Taylor%2527s_theorem/gxQTaNQF">Taylor's theorem</a>, for each z in the unit disk, there exists 0 ≤ t ≤ 1 such that f(z) = z + z2f″(tz) / 2. 
Thus, if |z| = 1/3 and |w| < 1/6, we have

|
 
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 f
 (
 z
 )
 −
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 −
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 z
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 1
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 f
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 1
 −
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 1
 6
 
 
 <
 
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 ≤
 
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 |
 
 .
 
 
 {\displaystyle |(f(z)-w)-(z-w)|={\frac {1}{2}}|z|^{2}|f''(tz)|\leq {\frac {|z|^{2}}{1-t|z|}}\leq {\frac {|z|^{2}}{1-|z|}}={\frac {1}{6}}<|z|-|w|\leq |z-w|.}

By <a href="/facts/Rouch%25C3%25A9%2527s_theorem/CNXqarau">Rouché's theorem</a>, the range of f contains the disk of radius 1/6 around 0.
Let D(z0, r) denote the open disk of radius r around z0. For an analytic function g : D(z0, r) → C such that g(z0) ≠ 0, the case above applied to (g(z0 + rz) − g(z0)) / (rg′(0)) implies that the range of g contains D(g(z0), |g′(0)|r / 6). 
For the general case, let f be an analytic function in the unit disk such that |f′(0)| = 1, and z0 = 0. 

<ul><li>If |f′(z)| ≤ 2|f′(z0)| for |z − z0| < 1/4, then by the first case, the range of f contains a disk of radius |f′(z0)| / 24 = 1/24.</li>
<li>Otherwise, there exists z1 such that |z1 − z0| < 1/4 and |f′(z1)| > 2|f′(z0)|.</li>
<li>If |f′(z)| ≤ 2|f′(z1)| for |z − z1| < 1/8, then by the first case, the range of f contains a disk of radius |f′(z1)| / 48 > |f′(z0)| / 24 = 1/24.</li>
<li>Otherwise, there exists z2 such that |z2 − z1| < 1/8 and |f′(z2)| > 2|f′(z1)|.</li></ul>
Repeating this argument, we either find a disk of radius at least 1/24 in the range of f, proving the theorem, or find an infinite sequence (zn) such that |zn − zn−1| < 1/2n+1 and |f′(zn)| > 2|f′(zn−1)|.
In the latter case the sequence is in D(0, 1/2), so f′ is unbounded in D(0, 1/2), a contradiction. 

<h3>Bloch's theorem</h3>
In the proof of Landau's Theorem above, Rouché's theorem implies that not only can we find a disk D of radius at least 1/24 in the range of f, but there is also a small disk D0 inside the unit disk such that for every w ∈ D there is a unique z ∈ D0 with f(z) = w. Thus, f is a <a href="/facts/Bijective/j4gTuTmW">bijective</a> analytic function from D0 ∩ f−1(D) to D, so its inverse φ is also analytic by the <a href="/facts/Inverse_function_theorem/pIwcyc6X">inverse function theorem</a>.

<h2 id="blochs-and-landaus-constants">Bloch's and Landau's constants</h2>
The number B is called the Bloch's constant. The lower bound 1/72 in Bloch's theorem is not the best possible. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown. 
The best known bounds for B at present are

0.4332
        ≈
        
          
            
              3
            
            4
          
        
        +
        2
        ×
        
          10
          
            −
            14
          
        
        ≤
        B
        ≤
        
          
            
              
                
                  
                    3
                  
                
                −
                1
              
              2
            
          
        
        ⋅
        
          
            
              Γ
              (
              
                
                  1
                  3
                
              
              )
              Γ
              (
              
                
                  11
                  12
                
              
              )
            
            
              Γ
              (
              
                
                  1
                  4
                
              
              )
            
          
        
        ≈
        0.47186
        ,
      
    
    {\displaystyle 0.4332\approx {\frac {\sqrt {3}}{4}}+2\times 10^{-14}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}\approx 0.47186,}

where Γ is the <a href="/facts/Gamma_function/SjGQcnyw">Gamma function</a>. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to <a href="/facts/Lars_Ahlfors/aukXlC2t">Ahlfors</a> and Grunsky.
The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown, but it is known that

0.5
 <
 L
 ≤
 
 
 
 Γ
 (
 
 
 1
 3
 
 
 )
 Γ
 (
 
 
 5
 6
 
 
 )
 
 
 Γ
 (
 
 
 1
 6
 
 
 )
 
 
 
 =
 0.543258965342...
 
 
 
 
 {\displaystyle 0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}=0.543258965342...\,\!}
 
 (sequence A081760 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)
In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.
For <a href="/facts/Injective/5ES6tqf5">injective</a> holomorphic functions on the unit disk, a constant A can similarly be defined. It is known that

0.5
 <
 A
 ≤
 0.7853
 
 
 {\displaystyle 0.5<A\leq 0.7853}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Mathematical_constant/rzqTX307">Table of selected mathematical constants</a></li></ul>

<ul><li><a href="/facts/Lars_Ahlfors/aukXlC2t">Ahlfors, Lars Valerian</a>; <a href="/facts/Helmut_Grunsky/MVsm31Ir">Grunsky, Helmut</a> (1937). "Über die Blochsche Konstante". <a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a>. 42 (1): 671–673. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01160101">10.1007/BF01160101</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122925005">122925005</a>.</li>
<li>Baernstein, Albert II; Vinson, Jade P. (1998). "Local minimality results related to the Bloch and Landau constants". Quasiconformal mappings and analysis. Ann Arbor: Springer, New York. pp. 55–89.</li>
<li>Bloch, André (1925). <a href="http://www.numdam.org/item/AFST_1925_3_17__1_0.pdf">"Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation"</a> (PDF). Annales de la Faculté des Sciences de Toulouse. 17 (3): 1–22. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Fafst.335">10.5802/afst.335</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0240-2963">0240-2963</a>.</li>
<li>Chen, Huaihui; Gauthier, Paul M. (1996). "On Bloch's constant". <a href="/facts/Journal_d%2527Analyse_Math%25C3%25A9matique/dQD4alPo">Journal d'Analyse Mathématique</a>. 69 (1): 275–291. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02787110">10.1007/BF02787110</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123739239">123739239</a>.</li>
<li>Landau, Edmund (1929), "Über die Blochsche Konstante und zwei verwandte Weltkonstanten", Mathematische Zeitschrift, 30 (1): 608–634, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01187791">10.1007/BF01187791</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120877278">120877278</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/BlochConstant.html">"Bloch Constant"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/LandauConstant.html">"Landau Constant"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

Bloch's theorem (complex analysis) open-in-new

Bloch's theorem (complex analysis)