CARATHÉODORY'S INEQUALITY ON THE BOUNDARY
CARATHÉODORY'S INEQUALITY ON THE BOUNDARY
The Pure and Applied Mathematics. 2015. May, 22(2): 169-178
• Received : January 30, 2015
• Accepted : May 01, 2015
• Published : May 31, 2015 PDF e-PUB PubReader PPT Export by style
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BüULENT NAFI, ÖORNEK

Abstract
In this paper, a boundary version of Carathéodory’s inequality is investigated. Also, new inequalities of the Carathéodory’s inequality at boundary are obtained and the sharpness of these inequalities is proved.
Keywords
1. INTRODUCTION
In recent years, boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz  , R. Osserman  , V. N. Dubinin  , M. Jeong [4 , 5] , H. P. Boas  and other’s studies. On the other hand, in the book  , Sharp Real-Parts Theorem’s (in particular Carathéodory’s inequalities), which are frequently used in the theory of entire functions and analytic function theory, have been studied.
The classical Schwarz lemma states that an holomorphic function f mapping the unit disc D = { z : | z | < 1} into itself, with f (0) = 0, satisfies the inequality | f ( z )| ≤ | z | for any point z D and | f '(0)| ≤ 1. Equality in these inequalities (in the first one, for z ≠ 0) occurs only if f ( z ) = λz , | λ | = 1 [3, p.329]. It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point z 0 with | z 0 | = 1, and if | f ( z 0 )| = 1 and f '( z 0 ) exists, then | f '( z 0 )| ≥ 1, which is known as the Schwarz lemma on the boundary.
In this paper, we studied “boundary Carathéodory’s inequalities” as analog the Schwarz lemma at the boundary  .
The Carathéodory’s inequality states that, if the function f is holomorphic on the unit disc D with f (0) = 0 and ℜ f A in D , then the inequality PPT Slide
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holds for all z D , and moreover PPT Slide
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Equality is achieved in (1.1) (for some nonzero z D ) or in (1.2) if and only if f ( z ) is the function of the form PPT Slide
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where θ is a real number [7, pp.3-4].
Robert Osserman considered the case that only one boundary fixed point of f is given and obtained a sharp estimate based on the values of the function. He has first showed that PPT Slide
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and PPT Slide
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under the assuumption f (0) = 0 where f is a holomorphic function mapping the unit disc into itself and z 0 is a boundary point to which f extends continuously and | f ( z 0 )| = 1. In addition, the equality in (1.3) holds if and only if f is of the form PPT Slide
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where θ is a real number and α D satisfies argα = arg z 0 . Also, the equality in (1.4) holds if and only if f ( z ) = ze , where θ is a real number.
Moreover, if PPT Slide
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then PPT Slide
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It follows that PPT Slide
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with equality only if f is of the form f ( z ) = zpe , θ real  .
If, in addition, the function f has an angular limit f ( z 0 ) at z 0 ∂D , | f ( z 0 )| = 1, then by the Julia-Wolff lemma the angular derivative f '( z 0 ) exists and 1 ≤ | f '( z 0 )| ≤ ∞ (see  ).
The inequality (1.5) is a particular case of a result due to Vladimir N. Dubinin in (see  ), who strengthened the inequality | f '( z 0 )| ≥ 1 by involving zeros of the function f . Some other types of strengthening inequalities are obtained in (see  ,  ).
We have following results, which can be offered as the boundary refinement of the Carathéodory’s inequality.
Theorem 1.1. Let f be a holomorphic function in the unit disc D , f (0) = 0 and PPT Slide
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for | z | < 1. Further assume that , for some z 0 ∂D , f has an angular limit f ( z 0 ) at z 0 , ℜf ( z 0 ) = A . Then PPT Slide
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Moreover , the equality in (1.7) holds if and if PPT Slide
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wehere θ is a real number .
Proof . The function PPT Slide
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is holomorphic in the unit disc D , | ϕ ( z )| < 1, ϕ (0) = 0 and | ϕ ( z 0 )| = 1 for z 0 ∂D .
That is,
| f ( z ) − 2 A | 2 = | f ( z )| 2 − 2ℜ ( f ( z )2 A ) + 4 A 2 = | f ( z )| 2 − 4 A ℜ ( f ( z )) + 4 A 2 .
From the hypothesis, since ℜ f ( z ) ≤ A and 4 A f ( z ) ≤ 4 A 2 , we take
|2 A f ( z )|2 ≥ | f ( z )| 2 − 4 A f ( z ) + 4 A f ( z ) = | f ( z )| 2 .
Therefore, we obtain PPT Slide
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From (1.4), we obtain PPT Slide
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So, we take PPT Slide
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If PPT Slide
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from (1.9) and | ϕ '( z 0 )| = 1, we obtain PPT Slide
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Theorem 1.2. Let f be a holomorphic function in the unit disc D , f (0) = 0 and PPT Slide
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for | z | < 1. Further assume that , for some z 0 ∂D , f has an angular limit f ( z 0 ) at z 0 , ℜ f ( z 0 ) = A . Then PPT Slide
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The inequality (1.10) is sharp , with equality for the function PPT Slide
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where PPT Slide
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is an arbitrary number on [0, 1] ( see (1.2)).
Proof . Using the inequality (1.3) for the function (1.8), we obtain PPT Slide
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and PPT Slide
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Now, we shall show that the inequality (1.10) is sharp. Choose arbitrary a ∈ [0, 1]
Let PPT Slide
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Then PPT Slide
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and PPT Slide
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Since | f '(0)| = 2A a , (1.10) is satisfied with equality.
An interesting special case of Theorem1.2 is when f ''(0) = 0, in which case inequality (1.10) implies PPT Slide
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Clearly equality holds for PPT Slide
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θ real.
Now, if PPT Slide
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is a holomorphic function in the unit disc D and PPT Slide
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for | z | < 1, it can be seen that Carathéodory’s inequality can be obtained with standard methods as follows: PPT Slide
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and PPT Slide
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The following result is a generalization of Theorem1.1.
Theorem 1.3. Let PPT Slide
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cp ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D and PPT Slide
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for | z | < 1. Further assume that, for some z 0 ∂D , f has an angular limit f ( z 0 ) at z 0 , ℜ f ( z 0 ) = A . Then PPT Slide
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In addition, the equality in (1.12) holds if and if PPT Slide
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wehere θ is a real number .
Proof . Using the inequality (1.6) for the function (1.8), we obtain PPT Slide
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Therefore, we take PPT Slide
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If PPT Slide
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from (1.13) and | ϕ ' ( z 0 )| = p , we obtain PPT Slide
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Theorem 1.4. Under hypotheses of Theorem 1.3, we have PPT Slide
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The inequality (1.14) is sharp , with equality for the function PPT Slide
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where PPT Slide
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is arbitrary number from [0, 1] ( see (1.11)).
Proof . Using the inequality (1.5) for the function (1.8), we obtain PPT Slide
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where PPT Slide
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Since PPT Slide
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we may write PPT Slide
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Thus, we take PPT Slide
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The equality in (1.14) is obtained for function PPT Slide
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as show simple calculations.
Consider the following product: PPT Slide
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B ( z ) is called a finite Blaschke product, where PPT Slide
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. Let the function PPT Slide
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satisfy the conditions of Carathéodory’s inequality and also have zeros a 1 , a 2 , ..., a n with order k 1 , k 2 , . . . , k n , respectively. Thus, one can see that Carathéodory’s inequality can be strengthened with the standard methods as follows: PPT Slide
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and PPT Slide
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The inequalities (1.15) and (1.16) show that the inequalities (1.1) and (1.2) will be able to be strengthened, if the zeros of function which are different from origin of f ( z ) in the (1.12) and (1.14) are taken into account.
Theorem 1.5. Let PPT Slide
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cp ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D , and PPT Slide
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for | z | < 1. Assume that for some z 0 ∂D , f has an angular limit f ( z 0 ) at z 0 , ℜ f ( z 0 ) = A . Let a 1 , a 2 , ..., an be zeros of the function f in D that are different from zero. Then we have the inequality PPT Slide
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In addition, the equality in (1.17) occurs for the function PPT Slide
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where a 1 , a 2 , ..., an are positive real numbers .
Proof . Let ϕ ( z ) be as in the proof of Theorem1.1 and a 1 , a 2 , ..., an be zeros of the function f in D that are different from zero. PPT Slide
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is a holomorphic functions in D , and | B ( z )| < 1 for | z | < 1. By the maximum principle for each z D , we have
| ϕ ( z )| ≤ | B ( z )| .
The auxiliary function PPT Slide
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is holomorphic in D , and | φ ( z )| < 1 for | z | < 1, φ (0) = 0 and | φ ( z 0 )| = 1 for z 0 ∂D .
Moreover, it can be seen that PPT Slide
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Besides, with the simple calculations, we take PPT Slide
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From (1.5), we obtain PPT Slide
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where PPT Slide
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Since PPT Slide
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we may write PPT Slide
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Therefore, we have PPT Slide
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Now, we shall show that the inequality (1.17) is sharp. Let PPT Slide
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Then PPT Slide
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and PPT Slide
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Since a 1 , a 2 , ..., an are positive real numbers, we take PPT Slide
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Since PPT Slide
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(1.17) is satisfied with equality.
References