On <i>q</i>-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

Raza, Mohsan; Naz, Hira; Malik, Sarfraz Nawaz; Islam, Sahidul

doi:https://doi.org/10.1155/2021/5353372

Journal of Mathematics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Special Issue

Methods in Dynamical Systems and Applications in Engineering

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 5353372 | https://doi.org/10.1155/2021/5353372

On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

Mohsan Raza,¹Hira Naz,²Sarfraz Nawaz Malik,²and Sahidul Islam³

Academic Editor: Fairouz Tchier

Received21 Sept 2021

Accepted15 Oct 2021

Published01 Nov 2021

Abstract

This article comprises the study of differential subordination with analogue of -derivative. It includes the sufficient condition on for to be subordinated by , , and implies that , where is the analytic function in the open unit disk. Moreover, certain sufficient conditions for -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

1. Introduction

Let a set be considered as the class of analytic functions defined in open unit disk under normalization conditions and , havingas Taylor series. The class comprises the normalized univalent functions, defined in . The major subcategories of class are of convex functions and of starlike functions. The class is another important class of analytic univalent functions whose co-domains are restricted to the right half plane and are used to determine the convexity and starlikeness of univalent functions. For more details, see [1, 2].

Let and be two analytic functions in . Then, is subordinated by , denoted as if can be written in the form of composition of and as subject to the existence of analytic function which satisfies the condition that and . Furthermore, if both and are univalent functions in , then implies that and .

Subordination plays an important role in univalent function theory, and this concept was first introduced by Lindelöf, but Littlewood [3, 4] contributed remarkably to this field. Differential subordination is actually the generalized version of differential inequalities with real variables. Many researchers contributed in the work related to differential subordinations. Historical developments in the field of differential subordination are briefly described by Miller and Mocanu in [5].

The advancement in the field of differential subordination starts with the usage of univalent functions. It was noticed in an article by Miller et al. [6] in 1974. Furthermore, many developments in this field have been achieved with the usage of differential subordinations in past fifty years. Differential inequality was a very well-known concept of real variables, and to study it in terms of complex variables, Miller and Mocanu [7] in 1981 were the first ones to introduce the idea of differential subordination. The contribution of Ruscheweyh and Singh [8] and Ruscheweyh and Wilken [9] is also of great importance in this field. Well-known Jack’s lemma [10] has brought the advancements in differential subordinations. Dziok [11] worked on some of the applications of Jack’s lemma. The research work carried out by Ma and Minda [12] in the function theory is worth to mention here, as they introduced the analytic function , which satisfies the conditions of normalization and having real part positive. The authors in [12] utilized the function and introduced the subclass of starlike functions as follows:

The idea presented in [12] is very useful, and it helped many researchers for further studies in this direction. Ali et al. [13, 14] worked on differential subordination for sufficiency criteria of Janowski starlikeness and evaluated several differential subordinations such as and found . Also, Ravichandran et al. [15] used this concept to find the sufficient conditions for starlikeness of Bernoulli’s lemniscate and Janowski functions. Sharma et al. [16] studied the differential subordinations to prove the starlikeness associated with cardioid domain and Halim et al. [17] introduced the concept for limacon domain.

Jackson [18, 19] was the one who introduced the -derivatives and -integrals. After Jackson, Srivastava was amongst the pioneers to contribute in the -calculus for its usage for analytic functions and their subclasses. Not only this, but he also applied -hypergeometric function in the functions theory. All these contributions are comprised in his book (pp. 347 in [20]). Ismail et al. [21] contributed in the -calculus for the study of starlike functions. Anastassiu and Gal [22, 23] also played their part in the development of complex variables with -generalization. Purohit et al. [24] have used fractional -calculus operators to apply subordination conditions on the class of non-Bazilevic functions. Sahoo and Agrawal [25] worked on starlike functions in -calculus and extended the idea of -starlikeness for particular subclasses of starlike functions. The involvement of -derivative in the class gave the formation of following subclass of starlike functions which was introduced by Aouf and Seoudy [26].

The class described above has drawn the attention of many researchers. Replacing with different functions such as Janowski, lemniscate of Bernoulli, cardioid, and limacon, the researchers got the new directions to the study. Srivastava et al. [27] studied -derivatives to find the relation between different classes of -starlike functions related to Janowski function. Srivastava et al. [28] introduced the class of -starlike functions by using general conic domains. They also obtained the bounds on Hankel and Toeplitz determinants for -starlike functions and continued working par excellence. They produced unmatchable results that worked as great motivation for many researchers worldwide. To have an idea of their remarkable work, one can see [29–31], [20, 27, 28, 32–43]. Contributions of Haq et al. [44] and Zainab et al. [45] are also worth to mention. They studied -analogue of differential subordinations for star-like functions related to limacon and cardioid domains, and Janowski functions. The -derivative is the foundation of all this work in -analogue, and it is defined as follows.

The -derivative of a complex valued function , defined in the domain , is given as follows:where . This implies the following:on the assumption that the function is differentiable in . The -derivative of an analytic function has Taylor series of the formwhere

For more details about -derivative and recent work on it, we refer the readers to [29–31], [20, 27, 28, 32–43]. In addition, the -analogue of Jack’s lemma has played a vital role in this paper which states as follows.

Lemma 1 (see [46]). Let be an analytic function in with . For maximum of on at , where and , then we havewhere with .

2. Main Results

Theorem 1. Assume that

Consider an analytic function on with which satisfies

Also, suppose

Here, is an analytic function in such that . Then, we have

Proof. Suppose thatwhere is analytic, and we have . Also, consider thatNow, we prove that , whereUsing (13) and (14), we obtainAlso, we haveConsider a point such thatNow, by using Lemma 1, we have . Now, consider that ; then, for , we obtainConsider a new functionThen,Above expression represents that the function has increasing behavior, so we have its minimum value at andSo, we conclude thatNow, from (9), we haveSince this result contradicts (10), therefore, , which completes the proof.
By taking we deduce the following result.

Corollary 1. Let and , satisfy the subordination

Then, .

Theorem 2. Assume that

Consider an analytic function on with which satisfies

Also, supposewhere is analytic function on with . Then,

Proof. We define a functionwhere is analytic and . Now, consider thatTo obtain the result, we have to show that . Using (30) and (31), we obtain the resultAlso, we haveConsider a point such thatNow, by using Lemma 1, we have . Now, consider that ; then, for , we obtainConsiderThen,Above expression represents that function has increasing behavior, so we have its minimum value at andSo, we conclude thatNow, from (26), we havewhich contradicts (27), and hence, , which completes the proof.
By taking we deduce the following result.

Corollary 2. Let and , which satisfies the subordination

Then, .

Theorem 3. Assume that

Consider an analytic function on with which satisfies

Also, supposewhere is analytic function on with . Then,

Proof. We define a functionwhere is analytic, and we have . Now, consider thatTo obtain the result, we have to show that . Using (46) and (47), we obtain the resultAlso, we haveConsider a point such thatNow, by using Lemma 1, we have . Now, consider that ; then, for , we obtainConsider a functionThen,Here, is clearly an increasing function, so we have its minimum value at andSo, we conclude thatNow, from (42), we havewhich contradict (43), and hence, , which completes the proof.
By taking we deduce the following result.

Corollary 3. Let and , satisfies the subordination

Then, .

Theorem 4. Assume that

Consider an analytic function on with which satisfies

Also, supposewhere is analytic function on with . Then,

Proof. We define a functionwhere is analytic, and we have . Now, consider thatTo obtain the result, we have to show that . Using (62) and (63), we obtainAlso, we haveConsider a point such thatNow, by using Lemma 1, we have . Now, consider that ; then, for , we obtainConsider a functionThen,Above expression represents that function has increasing behavior, so we have its minimum value at andSo, we conclude thatNow, from (58), we havewhich contradicts (59), and hence, , which completes the proof.
By taking we deduce the following result.

Corollary 5. Let and , satisfy the subordination

Then, .

Theorem 6. Assume that

Consider an analytic function on with which satisfies

Also, supposewhere is analytic function on with . Then,

Proof. We define a functionwhere is analytic, and we have . Now, consider thatTo obtain the result, we have to show that . Using (78) and (79), we obtainAlso, we haveConsider a point such thatNow, by using Lemma 1, we have . Now, consider that ; then, for , we obtainConsiderThen,Above expression represents that function has increasing behavior, so we have its minimum value at andSo, we conclude thatNow, from (74), we havewhich contradict (75), and hence, , which completes the proof.
By taking we deduce the following result.

Corollary 7. Let and , satisfy the subordination

Then, .

3. Conclusion

In this article, we have investigated the -differential subordination by using -version of well-known Jack’s Lemma. We have found the condition on such that implies that . These results have been utilized to find sufficient conditions for star-like functions related to lemniscate of Bernoulli. This method can further be applied to find sufficient conditions for star-like functions of Ma–Minda type.

Data Availability

No data were used in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and approved the final manuscript.

References

P. L. Duren, Univalent functions. Grundlehren der Math. Wissenchaften, Springer-Verlag, New York, NY, USA, 1983.
A. W. Goodman, “Univalent functions,” Mariner Publishing Company, vol. 1, no. 2, 1983.
View at: Google Scholar
J. E. Littlewood, Lectures on the Theory of Functions, Oxford University Press, London, UK, 1944.
J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathematical Society, vol. 2, no. 23, pp. 481–519, 2002.
View at: Google Scholar
S. S. Miller and P. T. Mocanu, Differential Subordinations Theory and Applications, Marcel Dekker, New York, NY, USA, 2000.
S. S. Miller, P. T. Mocanu, and M. O. Reade, “Bazilevic functions and generalized convexity,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 19, pp. 213–224, 1974.
View at: Google Scholar
S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” Michigan Mathematical Journal, vol. 28, pp. 157–171, 1981.
View at: Publisher Site | Google Scholar
S. Ruscheweyh and V. Singh, “On a Briot-Bouquet equation related to univalent functions,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 24, pp. 285–290, 1979.
View at: Google Scholar
S. Ruscheweyh and D. R. Wilken, “Sharp estimates for certain Briot-Bouquet subordinations,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 30, pp. 559–569, 1985.
View at: Google Scholar
I. S. Jack, “Functions starlike and convex of order α,” Journal of the London Mathematical Society, vol. 2-3, no. 3, pp. 469–474, 1971.
View at: Publisher Site | Google Scholar
D. Jacek, “Applications of the Jack lemma,” Acta Mathematica Hungaria, vol. 105, pp. 93–102, 2004.
View at: Google Scholar
W. C. Ma and D. A. Minda, “Unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis, pp. 157–169, International Press, Cambridge, MA, USA, 1992.
View at: Google Scholar
R. M. Ali, V. Ravichndaran, and N. Seenivasagan, “Sufficient conditions for janoswski starlikeness,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 9, p. 56, 2007.
View at: Google Scholar
R. M. Ali, N. E. Cho, V. Ravichandran, and S. S. Kumar, “Differential subordination for functions associated with the lemniscate of Bernoulli. Taiwan,” Jurnal Matematika, vol. 16, pp. 1017–1026, 2012.
View at: Publisher Site | Google Scholar
K. Sivaprasad, K. Virendra, V. Ravichandran, and E. C. Nak, “Sufficient conditions for starlike functions associated with the lemniscates of Bernoulli,” Journal of Inequalities and Applications, vol. 173, 2013.
View at: Google Scholar
K. Sharma, N. K. Jain, and V. Ravichandran, “Starlike functions associated with a cardioid,” Afrika Matematika, vol. 27, no. 5-6, pp. 923–939, 2016.
View at: Publisher Site | Google Scholar
Y. Yunus, S. A. Halim, and A. B. Akbarally, “Subclass of starlike functions associated with a limacon,” AIP Conference Proceedings, vol. 1974, Article ID 030023, 2018.
View at: Google Scholar
F. H. Jackson, “On -functions and certain difference operator,” Transactions of the Royal Society of Edinburgh, vol. 46, pp. 253–281, 1908.
View at: Google Scholar
F. H. Jackson, “On -definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910.
View at: Google Scholar
H. M. Srivastava, “Univalent functions, fractional calculus, and associated generalized hypergeometric functions in Univalent Functions,” in Fractional Calculus, and Their Applications, H. M. Srivastava and S. Owa, Eds., pp. 329–354, Wileya, J. and Sons, New York, NY, USA, 1989.
View at: Google Scholar
M. E. Ismail, E. Merkes, and D. Styer, “A generalization of Starlike functions,” Theory and Applications, vol. 14, no. 4, pp. 77–84, 1990.
View at: Publisher Site | Google Scholar
G. A. Anastassiu and S. G. Gal, “Geometric and approximation properties of some singular integrals in the unit disk,” Journal of Inequalities and Applications, vol. 1, Article ID 17231, p. 19, 2006.
View at: Google Scholar
G. A. Anastassiu and S. G. Gal, “Geometric and approximation properties of generalized singular integrals,” Journal of the Korean Mathematical Society, vol. 23, no. 2, pp. 425–443, 2006.
View at: Google Scholar
S. Abelman, K. A. Selvakumaran, M. M. Rashidi, and S. D. Purohit, “Subordination conditions for a class of non-bazilevič type defined by using fractional Q-calculus operators,” Facta Universitatis – Series: Mathematics and Informatics, vol. 32, no. 2, pp. 255–267, 2017.
View at: Publisher Site | Google Scholar
S. Agrawal and S. K. Sahoo, “A generalization of starlike functions of order alpha,” Hokkaido Mathematical Journal, vol. 46, pp. 15–27, 2017.
View at: Publisher Site | Google Scholar
T. M. Seoudy and M. K. Aouf, “Coefficient estimates of new classes of q-starlike and q-convex functions of complex order,” Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 135–145, 2016.
View at: Publisher Site | Google Scholar
H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, and N. Khan, “Some general classes of -starlike functions associated with the Janowski functions,” Symmetry Plus, vol. 11, Article ID 292, 2019.
View at: Google Scholar
H. M. Srivastava, M. Tahir, B. Khan, Q. Ahmad, and N. Khan, “Some general families of q-starlike functions associated with the Janowski functions,” Filomat, vol. 33, no. 9, pp. 2613–2626, 2019.
View at: Publisher Site | Google Scholar
B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, and Q. Z. Ahmad, “Coefficient estimates for a subclass of analytic functions associated with a certain leaf-like domain,” Mathematics, vol. 8, no. 8, p. 1334, 2020.
View at: Publisher Site | Google Scholar
S. Mahmood, M. Jabeen, S. N. Malik, H. M. Srivastava, R. Manzoor, and S. M. J. Riaz, “Some coefficient inequalities of -starlike functions associated with conic domain defined by -derivative,” Journal of Function Spaces, vol. 2018, Article ID 8492072, 13 pages, 2018.
View at: Google Scholar
M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus, and S. Kiran, “An upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with k-fibonacci numbers,” Symmetry Plus, vol. 12, no. 6, p. 1043, 2020.
View at: Publisher Site | Google Scholar
H. M. Srivastava, “Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis,” Iranian Journal of Science and Technology Transaction A-Science, vol. 44, no. 1, pp. 327–344, 2020.
View at: Publisher Site | Google Scholar
H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, and B. Khan, “Hankel and Toeplitz determinants for a subclass of -starlike functions associated with a general conic domain,” Mathematics, vol. 7, Article ID 181, 2019.
View at: Google Scholar
H. M. Srivastava, M. K. Aouf, and A. O. Mostafa, “Some properties of analytic functions associated with fractional q-calculus operators -calculus operators,” Miskolc Mathematical Notes, vol. 20, no. 2, pp. 1245–1260, 2019.
View at: Publisher Site | Google Scholar
H. M. Srivastava, M. Arif, M. Arif, and M. Raza, “Convolution properties of meromorphically harmonic functions defined by a generalized convolution $ q $-derivative operator,” AIMS Mathematics, vol. 6, no. 6, pp. 5869–5885, 2021.
View at: Publisher Site | Google Scholar
H. M. Srivastava, S. Arjika, and A. S. Kelil, “Some homogeneous -difference operators and the associated generalized Hahn polynomials,” Applied Set-Valued Analysis and Optimization, vol. 1, pp. 187–201, 2019.
View at: Google Scholar
H. M. Srivastava and D. Bansal, “Close-to-convexity of a certain family of -Mittag-Leffler functions,” Journal of Nonlinear and Variational Analysis, vol. 1, pp. 61–69, 2017.
View at: Google Scholar
H. M. Srivastava, J. Cao, and S. Arjika, “A note on generalized -difference equations and their applications involving -hypergeometric functions,” Symmetry Plus, vol. 12, Article ID 1816, 2020.
View at: Google Scholar
H. M. Srivastava and S. Arjika, “A general family of -hypergeometric polynomials and associated generating functions,” Mathematics, vol. 9, Article ID 1161, 2021.
View at: Google Scholar
H. M. Srivastava, B. Khan, N. Khan, and Q. Z. Ahmad, “Coefficient inequalities for -starlike functions associated with the Janowski functions,” Hokkaido Mathematical Journal, vol. 48, pp. 407–425, 2019.
View at: Publisher Site | Google Scholar
H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, and N. Khan, “Upper bound of the third Hankel determinant for a subclass of -starlike functions associated with the -exponential function,” Bulletin des Sciences Mathematiques, vol. 167, Article ID 102942, 2021.
View at: Google Scholar
H. M. Srivastava, S. Khan, S. Khan, Q. Z. Ahmad, N. Khan, and S. Hussain, “The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator,” Studia Universitatis Babes-Bolyai Matematica, vol. 63, no. 4, pp. 419–436, 2018.
View at: Publisher Site | Google Scholar
H. M. Srivastava, N. Raza, E. S. A. AbuJarad, G. Srivastava, and M. H. AbuJarad, “Fekete-Szegö inequality for classes of (p, q)-Starlike and (p, q)-convex functions,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 4, pp. 3563–3584, 2019.
View at: Publisher Site | Google Scholar
M. Ul-Haq, M. Raza, M. Arif, Q. Khan, and H. Tang, “Q-analogue of differential subordinations,” Mathematics, vol. 7, no. 8, p. 724, 2019.
View at: Publisher Site | Google Scholar
S. Zainab, A. Shakeel, M. Imran et al., “Sufficiency criteria for -starlike functions associated with cardioid,” Journal of Function Spaces, vol. 2021, Article ID 9999213, 9 pages, 2021.
View at: Publisher Site | Google Scholar
A. Çetinkaya and Y. Polatoglu, “-Harmonic mappings for which analytic part is -convex functions of complex order,” Hacettepe Journal of Mathematics and Statistics, vol. 47, pp. 813–820, 2018.
View at: Google Scholar

Copyright

Copyright © 2021 Mohsan Raza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

278

Downloads

444

Citations