A Connection between Basic Univalence Criteria

and Applied Analysis 3 Elementary calculation gives w (z, t) = ( f 󸀠 (e −t z) (1 + c) g (e −t z) − 1) e −2(α+β)t

Denote by A the class of analytic functions in U which satisfy the usual normalization (0) =   (0) − 1 = 0.
Furthermore, the first extension of univalence criteria was obtained by Pascu in [9].In his paper, starting from an analytic function  in the unit disk he established not only the univalence of  but also the analyticity and the univalence of a whole class of functions defined by an integral operator.
Other extensions of the univalence criteria, for an integral operator, were obtained in the papers [10][11][12][13][14]. From the main result of this paper, we found all the univalence criteria mentioned earlier and at the same time other new ones.

Loewner Chains
Before proving our main result we need a brief summary of theory of Loewner chains.
The following result due to Pommerenke is often used to obtain univalence criteria.

Main Result
Making use of Theorem 1, the essence of which is the construction of suitable Loewner chain, we can prove our main result.(2) are true for all  ∈ U \ {0}, then the function   , is analytic and univalent in U, where the principal branch is intended.

Specific Cases and Examples
Suitable choices of the functions  and ℎ and special values of the parameter  yield various types of univalence criteria.So, if in Theorem 2 we take  = 0 and ℎ() ≡ 0, we get the following result.then the function   defined by ( 5) is analytic and univalent in U.
Remark 5. Corollary 4 generalizes the well-known univalence criterion due to Becker.For  = 0 we found the result from [9].In the case when  = 0 and  = 1, the previous corollary reduces to Becker's criterion [7].
Remark 7. Corollary 6 represents a generalization of the univalence criterion due to Lewandowski.For  = 0 we found the result from [12].In the case when  = 0 and  = 1, the previous corollary reduces to Lewandowski's criterion [8].
For  =  and ℎ() ≡ 0, from Theorem 2 we can derived some results from paper [18].Theorem 8. Let  and  be complex numbers such that R ≥ 1/2, || < R( + ).For  ∈ A, if there exists an analytic function in U, () = 1 +  1  + ⋅ ⋅ ⋅ , such that the inequalities holds true for all  ∈ U, then the function   defined by ( 5) is analytic and univalent in U.
Proof.It is easy to check that inequality (28) implies inequality (26) of Theorem 8. Indeed, for | + 1| ≤ R( + ) and making use of ( 17), we have For the function () ≡ 1, from Theorem 8 we get the following.holds true for all  ∈ U, then the function   defined by ( 5) is analytic and univalent in U.In particular, the function  is univalent in U, where || < R(1 + ).
Remark 15.For special values of the parameters  and , from Corollary 14 we get some known results.For  = 0, we get the result given in [13].For  = 1, since  1 () = (), Corollary 14 generalizes the criterion of univalence due to Nehari, and for  = 1 and  = 0 we obtain the univalence criterion due to Nehari [4].
Remark 19.Corollary 18 represents a generalization of the univalence criterion due to Ozaki and Nunokawa.For  = 0 we found the result from [14].In the case when  = 0 and  = 1, Corollary 18 reduces to the univalence criterion of Ozaki and Nunokawa [6].
It follows that all the conditions of Corollary 18 are satisfied, and therefore the function   defined by ( 5) is analytic and univalent in U.
Remark 20.Theorem 2 gives us a connection between Alexander's theorem, Noshiro-Warschawski's theorem, and the univalence criteria of Becker, Lewandowski, Nehari, Goluzin, and Ozaki and Nunokawa as well as their generalizations.