Extension of Nunokawa Lemma for Functions with Fixed Second Coefficient and Its Applications

Abstract

In this paper, we study some properties of analytic functions with fixed initial coefficients. The methodology of differential subordination is used for modification and improvements of several well-known results for subclasses of univalent functions by restricting the functions with fixed initial coefficients. Actually, by extending the Nunokawa lemma for fixed initial coefficient functions, we obtain some novel results on subclasses of univalent functions, such as differential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new sufficient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with fixed second coefficients.

1. Introduction and Preliminaries

Let be the class of analytic functions in the unit disc . For and , let us define two well-known classes of analytic functions as follows:

We denote by and , as the class of univalent functions. Also, we denote by , the set of starlike function of order (), as follows:which is introduced by Robertson in [1]. Let be the classes of starlike functions. Brannan and Kirwan in [2] introduced the class of strongly starlike function of order , , by

Also, Takahashi and Nunokawa in [3] defined the following subclasses of :

It is easy to see that . Also, if , then . Then, they are the subclasses of univalent functions in the unit disc .

Recently, Ali et al. [4] have extended the theory of second-order differential subordination for functions with a fixed initial coefficient. We denote by the class of analytic functions with a fixed initial coefficient as follows:

Further, letand also, letwhere is fixed. We denote by , and we assume that and are positive real numbers.

The importance of the second coefficient of analytic functions was shown in the monograph [5], for example, coefficient estimates in two well-known growth and distortion theorems for functions in the class . On the other hand, one of the significant tools in geometric function theory is the theory of differential subordination due to Miller and Mocanu [6]. Recently, Ali et al. [4] improved the theory of differential subordination by this assumption that the second coefficient of analytic function is fixed. For some applications of this improvement, see [7, 8]. In addition, Nunokawa has proved a theorem known as Nunokawa lemmas [9] which have had many applications in the geometric function theory. So, it is natural to ask if it is possible to extend this lemma for analytic functions with a fixed second coefficient.

In this paper, we extend this lemma, and then, it will be applied to obtain several new results by restricting the functions to have a fixed second coefficient. It is a remarkable fact when we restrict ourselves to the functions with a fixed second coefficient; then, it will expand the validity domain of the results. As a byproduct of this idea, one can also get results for other class mapping. Even though, in this paper, we are not going into details of all such applications, we remark that other applications will appear in other papers of the authors.

In [10], Silverman examined the class , consisting of normalized functions that satisfy the conditionfor some positive . He has proved that .

For a fixed coefficient , let

For and in , we say that is subordinate to if there exists a function with , such that denote by . When is univalent in then we can see

This article is arranged as follows. In Section 2, we extend the lemma due to Nunokawa [9] for fixed second coefficient functions and then will bring its nice applications for finding sufficient conditions for strongly starlike functions. In Section 3, by considering the special cases of this extension, we will provide some sufficient conditions for starlikeness and univalency of analytic functions. Also, we discuss about preserving starlikeness of the general integral operator . Finally, in Theorem 26, we state a new result which is improvement of some well-known results in the literature.

We just need a definition and a fundamental lemma due to Ali et al. [4] to get the results.

Definition 1 ([6], Definition 1, p. 158). Let be the class of functions that are analytic and injective in , whereand are such that for .

Lemma 2 (see [4]). Let with , and with . If there exist a point such that and , thenwhere ( is real) and

2. Main Results

The Nunokawa lemma is a very useful tool in the theory of differential subordination that was proved for the first time by Nunokawa [9]. At first, we mention extension of this lemma with its proof because we need it in the next theorems.

Theorem 3. Let and in . If there exist and such that and for with , and is a real number with . Then, we havewhere

Proof. Let us definewith . It is easy to check that is analytic in with andAlso, with . According to properties of and assumption (14), we have , , and . SetThen, it can be verified that is the right half plane , with , with and . We notice thatand so, by considering , we can rewrite the functions and asNow, taking the derivative of and calculating the inverse of yieldsSince withand so, from Lemma 2, we deduce that there exist two complex numbers and on with and such thatwhereButhence,But the functiontakes its minimum at the point , and we have . In view of and (25), we obtainthus, we showed thatIn a similar way, by noting that with and lettingwe can see thatNow, doing analogous above, we find thatThus, we showed thatwhich is (16); therefore, the proof is complete.☐

Remark 4. Nunokawa [9] provided a similar result to Theorem 3 with this assumption that . But, if we fix the second initial coefficient, then we may have some reform as seen above.

Theorem 5. Let , and. If withwhere andthen .

Proof. Suppose that Let us definewithThen, it is clear that withAlso, it is easy to see thatIf , then is not contained in the right half plane ; hence, there exists a point such that is contained in the right half plane while, , or , where . In the first step, let , then by applying the same argument as Theorem 3, we obtainwhereAlso, from the relation of and , we have , and so, by making use of (41), it yieldsIf we definethen one can verify that takes its minimum at ; hence, applying (42) and (43), we havewhich contradicts with the assumption .
If , where , using a similar argument as above leads the contradiction. Indeed, applying the previous considerations, we obtainwhereNotice that , and so, by making use of (46), it yieldsIf we definethen by some calculation, we find that takes its minimum at ; hence, applying (47) and (48), we havewhich contradicts with the assumption . So, in both cases, we come to a contradiction, and the proof is complete.☐

Letting in the Theorem 5, we get the following corollary.

Corollary 6. Let and . If withthen .

Putting and in Corollary 6, we obtain the following result.

Corollary 7. If , andthen is strongly starlike of order .

In the following theorem, we mention a strong result which provides sufficient conditions to strongly starlike functions.

Theorem 8. Let , and . If andwhereand are defined as Theorem 5. Then,

Proof. Sincewe haveNow, for , we havewith , and for with . Hence, is a whole complex plane except the half lines and . Suppose thatwithThen, it is clear that withAlso, it is easy to see thatIf , then is not contained in the right half plane ; hence, there exists a point such that is contained in the right half plane while , or , where . In the first step, let ; then, using the same argument as Theorem 3, we obtainwhereAlso, from the relation of and , we haveand so, by making use of (64), it yieldswhich contradicts with the assumption.
For the case , where , using the same argument as Theorem 3, we obtainwhereNotice thatand so, by making use of (68), we havewhich contradicts with the assumption. So, in both cases, we come to a contradiction, and the proof is complete.☐

3. Starlikeness of Analytic Functions

There are many differential inequalities in geometric function theory which is used for univalency or starlikeness of analytic functions. For example, Ozaki [11] has proved that if satisfies the condition , then is univalent. Also, the well-known result known as the Mark-strohhäcker [5] states that , then

Hence, it is natural to extend the similar results to analytic functions with a fixed initial coefficient.

In this section, we try to obtain some inequalities in analytic functions with second fixed coefficients which improve earlier results obtained in the literature. At first, we bring the following corollary.

Corollary 9. Let and such that for and with . Then, we have

Proof. By letting in Theorem 3, we obtain (73). For (74), we see that is purely imaginary so and . Let us putThen, we have that with . From the assumption and Lemma 2, there exist with such thatwhich shows thatand is a negative real number and (74) is obtained.☐

Remark 10. The special case when reduces to the result due to Nunokawa [12].

Corollary 11. Let and in . If there exist such that for and , where . Then, we havewhen andwhen , where .

Proof. Let . We have for and . Let with . From Corollary 9, we havewhereSo, the proof is done.☐

Remark 12. Note the special case of Corollary 11 when reduces to the well-known lemma [9].

Theorem 13. Let , and suppose thatThen, we have .

Proof. From the assumption (82), we have . In fact, if has a zero of order at , then we havewhere is analytic in and . Then, we haveBut is fixed, so we haveThen, the imaginary part of the right-hand side of (84) can take any values when approaches . This contradicts (82). This shows that for . Therefore, if there exists a point such that , then we have . From Corollary 9, we obtainThis contradicts (82). This completes our proof.☐

Remark 14. The special case when reduces to the result obtained in [9].

Corollary 15. Let , and suppose thatThen, we have for .

Proof. Let and put . Then, . Applying Theorem 13, we get the result.☐

Corollary 16. Let and suppose thatThen, we have for .

Proof. Let and put . Then, . Applying Theorem 13, we get the result.☐

Corollary 17. Let , and suppose thatThen, we have for .

Proof. Let and put . Then, . Applying Theorem 13, we get the result.☐

Theorem 18. Let and in . Also, suppose that and . If there exist such that for and but where . Then, we havewhere

Proof. Letwhere . We know that is analytic and univalent in with and . So, with . From Lemma 2, there exist on with such thatWe havewhereBy taking for a fixed real , we haveIf , then for and for all real number , we have which leads toAlso, we have when , and is an arbitrary real number. This leads toTherefore, the proof is complete.☐

Corollary 19. Let and in. Suppose also that , andwhereThen, we have for .

Proof. If there exists a point, , , such thatThen, from Theorem 18, we haveand so,This contradicts (99), and the proof is complete.☐

By taking , and in Corollary 19, we obtain the following theorem.

Theorem 20. Let , and . IfwhereThen,

Remark 21. We remark that the special case of Theorem 20 with , and was proved in [8]. Also, the special case when , and reduces to the well-known Mark-strohhäcker result [13].

Also, by taking , and in Corollary 19, we obtain the following theorem.

Theorem 22. Let , , and . IfwhereThen,For let be the integral operator defined asWe note that the integral operator maps to Now, we state the next following theorems.

Theorem 23. Let , , and . If andwhereThen,where is defined in (110).

Proof. Let us define . It is easy to show thatNow considering Corollary 19, we get our result.☐

Putting , , and in the above theorem, we have the following result.

Corollary 24. Let . IfThen,By considering Corollary 19, we have the following result.

Theorem 25. Let , , and. If andwhereThen,where is defined in (110).

Finally, we state the following theorem which is very interesting and in the especial case improves some earlier results in the literature.

Theorem 26. Suppose that and . Also, letwhereIf is analytic in and omits , then is starlike in the unit disc and

Proof. Let us define the function asThen, it is easy to see that andIf there exists a point such that for and . Then, using Corollary 9, we havewhere is a real number with For the case and with , we obtainTherefore, we obtainand that contradicts the hypothesis of the theorem. In the case and with , by applying the same method as above, we also obtainand this also contradicts with the assumption of the theorem. Hence, we obtainBut the inequalityholds true if only ifor equivalentlyFinally, we note that the last inequality holds true if and only if , and this is proved in the above. Therefore, from (130), we haveand the proof is complete.☐

Putting in Theorem 26, we obtain the following result.

Corollary 27. Suppose that andIf is analytic in and omits , then is starlike in the unit disc and

Remark 28. Note that Corollary 27 improves the result of Sumit and Ravichandran [8].

Data Availability

In this paper, we do not use any data from elsewhere, since this paper does not need data.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.