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. In recent years, boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz [6] , R. Osserman [8] , V. N. Dubinin [2] , M. Jeong [4 , 5] , H. P. Boas [1] and other’s studies. On the other hand, in the book [7] , 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 [8] . 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 holds for all z ∈ D , and moreover 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 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 and 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 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^iθ , where θ is a real number. Moreover, if then It follows that with equality only if f is of the form f ( z ) = z^pe^iθ , θ real [8] . 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 [11] ). The inequality (1.5) is a particular case of a result due to Vladimir N. Dubinin in (see [2] ), 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 [9] , [10] ). 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 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 Moreover , the equality in (1.7) holds if and if wehere θ is a real number . Proof . The function 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 ℜ f ( z ) ≤ 4 A ² , we take |2 A − f ( z )|2 ≥ | f ( z )| ² − 4 A ℜ f ( z ) + 4 A ℜ f ( z ) = | f ( z )| ² . Therefore, we obtain From (1.4), we obtain So, we take If from (1.9) and | ϕ '( z ₀ )| = 1, we obtain Theorem 1.2. Let f be a holomorphic function in the unit disc D , f (0) = 0 and 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 The inequality (1.10) is sharp , with equality for the function where is an arbitrary number on [0, 1] ( see (1.2)). Proof . Using the inequality (1.3) for the function (1.8), we obtain and Now, we shall show that the inequality (1.10) is sharp. Choose arbitrary a ∈ [0, 1] Let Then and 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 Clearly equality holds for θ real. Now, if is a holomorphic function in the unit disc D and for | z | < 1, it can be seen that Carathéodory’s inequality can be obtained with standard methods as follows: and The following result is a generalization of Theorem1.1. Theorem 1.3. Let c_p ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D and 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 In addition, the equality in (1.12) holds if and if wehere θ is a real number . Proof . Using the inequality (1.6) for the function (1.8), we obtain Therefore, we take If from (1.13) and | ϕ ' ( z ₀ )| = p , we obtain Theorem 1.4. Under hypotheses of Theorem 1.3, we have The inequality (1.14) is sharp , with equality for the function where is arbitrary number from [0, 1] ( see (1.11)). Proof . Using the inequality (1.5) for the function (1.8), we obtain where Since we may write Thus, we take The equality in (1.14) is obtained for function as show simple calculations. Consider the following product: B ( z ) is called a finite Blaschke product, where . Let the function 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: and 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 c_p ≠ 0, p ≥ 1 be a holomorphic function in the unit disc D , and 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 , ..., a_n be zeros of the function f in D that are different from zero. Then we have the inequality In addition, the equality in (1.17) occurs for the function where a ₁ , a ₂ , ..., a_n are positive real numbers . Proof . Let ϕ ( z ) be as in the proof of Theorem1.1 and a ₁ , a ₂ , ..., a_n be zeros of the function f in D that are different from zero. 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 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 Besides, with the simple calculations, we take From (1.5), we obtain where Since we may write Therefore, we have Now, we shall show that the inequality (1.17) is sharp. Let Then and Since a ₁ , a ₂ , ..., a_n are positive real numbers, we take Since (1.17) is satisfied with equality.