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Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis

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Abstract

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are known to have widespread applications, particularly in several areas of number theory and combinatorial analysis such as (for example) the theory of partitions. Our usages here, in this survey-cum-expository article, of the q-calculus and the fractional q-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics. By applying a fractional q-calculus operator, we define the subclasses \({\mathcal{S}}_{n}^{\alpha }(\lambda ,\beta ,b,q)\) and \({\mathcal{G}}_{n}^{\alpha }(\lambda ,\beta ,b,q)\) of normalized analytic functions with complex order and negative coefficients. Among the results investigated for each of these function classes, we derive their associated coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points and growth and distortion theorems. Our investigation here is motivated essentially by the fact that basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several areas of number theory such as the theory of partitions. In fact, basic (or q-) hypergeometric functions are useful also in a wide variety of fields including, for example, combinatorial analysis, finite vector spaces, lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology and statistics (see also (Srivastava and Karlsson in Multiple Gaussian hypergeometric series. pp 350–351, 1985) and the references cited thereon). In the last section on conclusion, we choose to point out the fact that the results for the q-analogues, which we consider in this article for \(0< q < 1\), can easily (and possibly trivially) be translated into the corresponding results for the (p, q)-analogues (with \(0< q < p \leqq 1\)) by applying some obvious parametric and argument variations, the additional parameter p being redundant. Several other families of such extensively- and widely-investigated linear convolution operators as (for example) the Dziok–Srivastava, Srivastava–Wright and Srivastava–Attiya linear convolution operators, together with their extended and generalized versions, are also briefly considered.

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References

  • Abelomen S, Selvakumaran KA, Rashidi MM, Purohit SD (2017) Subordination conditions for a class of non-Bazilavič type defined by using fractional \(q\)-culculus operators. Facta Univ Ser Math Inf 32:255–267

    Google Scholar 

  • Abu-Risha MH, Annaby MH, Ismail ME-H, Mansour S (2007) Linear \(q-\)difference equations. Z Anal Anwend 26:481–494

    MathSciNet  MATH  Google Scholar 

  • Aghalary R, Joshi SB, Mohapatra RN, Ravichandran V (2007) Subordinations for analytic functions defined by the Dziok–Srivastava linear operator. Appl Math Comput 187:13–19

    MathSciNet  MATH  Google Scholar 

  • Agrawal S, Sahoo SK (2017) A generalization of starlike functions of order \(\alpha \). Hokkaido Math J 46:15–27

    MathSciNet  MATH  Google Scholar 

  • Altintaş O, Özkan Ö (2000) Neighborhoods of a class of analytic functions with negative coefficients. Appl Math Lett 13(3):63–67

    MathSciNet  MATH  Google Scholar 

  • Annaby MH, Mansour ZS (2012) \(q\)-Fractional calculus and equations. Springer, New York

    MATH  Google Scholar 

  • Aouf MK (2009) Neighborhoods of a certain family of multivalent functions defined by using a fractional derivative operator. Bull Belg Math Soc Simon Stevin 16:31–40

    MathSciNet  MATH  Google Scholar 

  • Aouf MK, Dziok J (2008) Distortion and convolutional theorems for operators of generalized fractional calculus involving Wright function. J Appl Anal 14:183–192

    MathSciNet  MATH  Google Scholar 

  • Aouf MK, Darwish HE, Sălăgean GS (2001) On a generalization of starlike functions with negative coefficients. Mathematica (Cluj) 66(43):3–10

    MathSciNet  MATH  Google Scholar 

  • Attiya AA, Yassen MF (2016) Majorization for some classes of analytic functions associated with the Srivastava–Attiya operator. Filomat 30:2067–2074

    MathSciNet  MATH  Google Scholar 

  • Cang Y-L, Liu J-L (2017) Some subclasses of meromorphically multivalent functions associated with the Dziok–Srivastava operator. Filomat 31:2449–2458

    MathSciNet  Google Scholar 

  • Cang Y-L, Liu J-L (2018) A family of multivalent analytic functions associated with Srivastava–Tomovski generalization of the Mittag–Leffler function. Filomat 32:4619–4625

    MathSciNet  Google Scholar 

  • Challab KA, Darus M, Ghanim F (2019) Some application on Hurwitz–Lerch zeta function defined by a generalization of the Srivastava–Attiya operator. Kraguj J Math 43:201–217

    MathSciNet  Google Scholar 

  • Chandrashekar R, Murugusundaramoorthy G, Lee SK, Subramanian KG (2009) A class of complex-valued harmonic functions defined by Dziok–Srivastava operator. Chamchuri J Math 1(2):31–42

    MathSciNet  MATH  Google Scholar 

  • Cho NE, Kim IH, Srivastava HM (2010) Sandwich-type theorems for multivalent functions associated with the Srivastava–Attiya operator. Appl Math Comput 217:918–928

    MathSciNet  MATH  Google Scholar 

  • Cho NE, Kwon OS, Ali RM, Ravichandran V (2011) Subordination and superordination for multivalent functions associated with the Dziok–Srivastava operator. J Inequal Appl 2011:1–17

    MathSciNet  MATH  Google Scholar 

  • Devi S, Srivastava HM, Swaminathan A (2016) Inclusion properties of a class of functions involving the Dziok–Srivastava operator. Korean J Math 24:139–168

    MathSciNet  Google Scholar 

  • Duren PL (1983) Univalent functions, grundlehren der mathematischen wissenschaften, vol 259. Springer, New York

    Google Scholar 

  • Dweby HA, Darus M (2013) A subclass of harmonic univalent functions associated with \(q\)-analogue of Dziok–Srivastava operator. ISRN Math Anal 2013:1–6

    MathSciNet  Google Scholar 

  • Dziok J (2007) On the convex combination of the Dziok–Srivastava operator. Appl Math Comput 188:1214–1220

    MathSciNet  MATH  Google Scholar 

  • Dziok J (2015) Generalizations of multivalent Mocanu functions. Appl Math Comput 269:965–971

    MathSciNet  MATH  Google Scholar 

  • Dziok J, Srivastava HM (1999) Classes of analytic functions associated with the generalized hypergeometric function. Appl Math Comput 103:1–13

    MathSciNet  MATH  Google Scholar 

  • Dziok J, Srivastava HM (2003) Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec Funct 14:7–18

    MathSciNet  MATH  Google Scholar 

  • El-Ashwah RM, Aouf MK, Shamandy A, El-Deeb SM (2013) A class of harmonic starlike functions with respect to Symmetric points associated with the Srivastava–Wright generalized hypergeometric function. J Class Anal 2:73–83

    MathSciNet  MATH  Google Scholar 

  • Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1953) Higher transcendental functions, vol I. McGraw-Hill Book Company, New York

    MATH  Google Scholar 

  • Ezhilarasi R, Sudharsan TV, Subramanian KG, Joshi SB (2012) A subclass of harmonic univalent functions with positive coefficients defined by Dziok–Srivastava operator. Gen Math 20(2–3):32–46

    Google Scholar 

  • Gasper G, Rahman M (1990) Basic hypergeometric series (with a Foreword by Richard Askey). Encyclopedia of mathematics and its applications, vol 35. Cambridge University Press, Cambridge

  • Gasper G, Rahman M (2004) Basic hypergeometric series. Encyclopedia of mathematics and its applications, vol 96, 2nd edn. Cambridge University Press, Cambridge

  • Goodman AW (1983) Univalent functions, vol I and II. Mariner Publishing Company Incorporated, Tampa

    MATH  Google Scholar 

  • Ibrahim RW, Darus M (2010) Subordination and superordination for functions based on Dziok–Srivastava linear operator. Bull Math Anal Appl 2(3):15–26

    MathSciNet  MATH  Google Scholar 

  • Ismail ME-H, Merkes E, Styer D (1990) A generalization of starlike functions. Complex Var Theory Appl 14:77–84

    MathSciNet  MATH  Google Scholar 

  • Jackson FH (1908) On \(q\)-functions and a certain difference operator. Trans R Soc Edinb 46:64–72

    Google Scholar 

  • Jackson FH (1910a) On \(q\)-definite integrals. Quart J Pure Appl Math 41:193–203

    MATH  Google Scholar 

  • Jackson FH (1910b) \(q\)-Difference equations. Am J Math 32:305–314

    MATH  Google Scholar 

  • Janowski W (1973) Some extremal problems for certain families of analytic functions. Ann Polon Math 28:297–326

    MathSciNet  MATH  Google Scholar 

  • Kanas S, Srivastava HM (2000) Linear operators associated with \(k\)-uniformly convex functions. Integral Transforms Spec Funct 9:121–132

    MathSciNet  MATH  Google Scholar 

  • Kanas S, Wiśniowska A (1999) Conic regions and \(k\)-uniform convexity. J Comput Appl Math 105:327–336

    MathSciNet  MATH  Google Scholar 

  • Kanas S, Wiśniowska A (2000) Conic domains and starlike functions. Rev Roum Math Pures Appl 45:647–657

    MathSciNet  MATH  Google Scholar 

  • Khan Q, Arif M, Raza M, Srivastava G, Tang H, Rehman SU, Ahmad B (2019) Some applications of a new integral operator in \(q\)-analog for multivalent functions. Mathematics 7:1–13

    Google Scholar 

  • Kiryakova VS (2011) Criteria for univalence of the Dziok–Srivastava and the Srivastava–Wright operators in the class \(A\). Appl Math Comput 218:883–892

    MathSciNet  MATH  Google Scholar 

  • Kwon OS, Cho NE (2007) Inclusion properties for certain subclasses of analytic functions associated with the Dziok–Srivastava operator. J Inequal Appl 2007:1–10

    MathSciNet  MATH  Google Scholar 

  • Li S-H, Ma L-N, Tang H (2015) Certain subclasses of analytic functions of complex order involving the generalized Dziok–Srivastava operator. Int J Math Anal 9:277–292

    Google Scholar 

  • Liu J-L (2019) New applications of the Srivastava–Tomovski generalization of the Mittag–Leffler function. Iran J Sci Technol Trans A Sci 43:519–524

    MathSciNet  Google Scholar 

  • Liu J-L, Srivastava HM (2004) Certain properties of the Dziok–Srivastava operator. Appl Math Comput 159:485–493

    MathSciNet  MATH  Google Scholar 

  • Liu Z-H, Wang Z-G, Wen F-H, Yong Sun (2012) Some subclasses of analytic functions involving the generalized Srivastava–Attiya operator. Hacet J Math Stat 41:421–434

    MathSciNet  MATH  Google Scholar 

  • Mahmood S, Jabeen M, Malik SN, Srivastava HM, Manzoor R, Riaz SMJ (2018) Some coefficient inequalities of \(q\)-starlike functions associated with conic domain defined by \(q\)-derivative. J Funct Spaces 2018:1–13

    MathSciNet  MATH  Google Scholar 

  • Mahmood S, Ahmad QZ, Srivastava HM, Khan N, Khan B, Tahir M (2019a) A certain subclass of meromorphically \(q\)-starlike functions associated with the Janowski functions. J Inequal Appl 2019:1–11

    MathSciNet  Google Scholar 

  • Mahmood S, Raza N, Abujarad ESA, Srivastava G, Srivastava HM, Malik SN (2019b) Geometric properties of certain classes of analytic functions associated with a \(q\)-integral operator. Symmetry 11:1–14

    MATH  Google Scholar 

  • Mahmood S, Srivastava HM, Khan N, Ahmad QZ, Khan B, Ali I (2019c) Upper bound of the third Hankel determinant for a subclass of \(q\)-starlike functions. Symmetry 11:1–13

    MATH  Google Scholar 

  • Miller SS, Mocanu PT (2000) Differential subordination: theory and applications, vol 225. Series on monographs and textbooks in pure and applied mathematics. Marcel Dekker Incorporated, New York and Basel

    Google Scholar 

  • Murugusundaramoorthy G, Thilagavathi K, Magesh N (2013) Majorization for certain classes of analytic functions of complex order associated with the Dziok–Srivastava and the Srivastava-Wright convolution operators. Tamsui Oxford J Inf Math Sci 29:477–486

    MathSciNet  MATH  Google Scholar 

  • Noor KI, Malik SN (2011) On coefficient inequalities of functions associated with conic domains. Comput Math Appl 62:2209–2217

    MathSciNet  MATH  Google Scholar 

  • Noor KI, Shah SA (2019) Application of Srivastava–Attiya operator to the generalization of Mocanu functions. Int J Anal Appl 17:674–685

    MATH  Google Scholar 

  • Omar R, Halim SA (2012) Multivalent harmonic functions defined by Dziok–Srivastava operator. Bull Malays Math Sci Soc (Ser 2) 35:601–610

    MathSciNet  MATH  Google Scholar 

  • Oros GI, Oros G, Kim IH, Cho NE (2011) Differential subordinations associated with the Dziok–Srivastava operator. Math Rep 13(1):57–64

    MathSciNet  MATH  Google Scholar 

  • Owa S, Srivastava HM (1987) Univalent and starlike generalized hypergeometric functions. Can J Math 39:1057–1077

    MathSciNet  MATH  Google Scholar 

  • Owa S, Murugusundaramoorthy G, Magesh N (2012) Subclasses of starlike functions involving Srivastava–Attiya integral operator. Bull Math Anal Appl 4(4):35–44

    MathSciNet  MATH  Google Scholar 

  • Piejko K, Sokół J (2006) On the Dziok–Srivastava operator under multivalent analytic functions. Appl Math Comput 77:839–843

    MathSciNet  MATH  Google Scholar 

  • Ponnusamy S, Singh V (1997) Criteria for strongly starlike functions. Complex Var Theory Appl 34:267–291

    MathSciNet  MATH  Google Scholar 

  • Purohit SD, Raina RK (2011) Certain subclasses of analytic functions associated with fractional calculus operators. Math Scand 109:55–70

    MathSciNet  MATH  Google Scholar 

  • Purohit SD, Raina RK (2013) Fractional \(q\)-calculus and certain subclasses of univalent analytic functions. Mathematica (Cluj) 55(78):62–74

    MathSciNet  MATH  Google Scholar 

  • Rehman MSU, Ahmad QZ, Srivastava HM, Khan B, Khan N (2020) Partial sums of generalized \(q\)-Mittag–Leffler functions. AIMS Math 5:408–420

    Google Scholar 

  • Selvakurmaran KA, Purohit SD, Secer A, Bayram M (2014) Convexity of certain \(q\)-integral operators of \(p\)-valent functions. Abstr Appl Anal 2014:1–7

    MathSciNet  MATH  Google Scholar 

  • Seoudy TM, Aouf MK (2014) Convolution properties for certain classes of analytic functions defined by \(q\)-derivative operator. Abstr Appl Anal 2014:1–7

    MathSciNet  MATH  Google Scholar 

  • Seoudy TM, Aouf MK (2016) Coefficient estimates of new class of \(q\)-starlike and \(q\)-convex functions of complex order. J Math Inequal 10:135–145

    MathSciNet  MATH  Google Scholar 

  • Shanmugam TN, Sivasubramanian S, Owa S (2007) Argument estimates of certain multivalent functions involving Dziok–Srivastava operator. Gen Math 15(4):83–100

    MathSciNet  MATH  Google Scholar 

  • Shi L, Khan Q, Srivastava G, Liu J-L, Arif M (2019) A study of multivalent \(q\)-starlike functions connected with circular domain. Mathematics 7:1–12

    Google Scholar 

  • Silverman H (1975) Univalent functions with negative coefficients. Proc Am Math Soc 51:109–116

    MathSciNet  MATH  Google Scholar 

  • Sim YJ, Kwon OS, Cho NE, Srivastava HM (2020) Bounds for the real parts and arguments of normalized analytic functions defined by the Srivastava–Attiya operator. J Comput Anal Appl 28:628–645

    Google Scholar 

  • Srivastava HM (1989) Univalent functions, fractional calculus, and associated generalized hypergeometric functions. In: Srivastava HM, Owa S (eds) Univalent functions, fractional calculus, and their applications. Wiley, New York, pp 329–354

    Google Scholar 

  • Srivastava HM (2007) Some Fox–Wright generalized hypergeometric functions and associated families of convolution operators. Appl Anal Discrete Math 1(1):56–71

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM (2014) A new family of the \(\lambda \)-generalized Hurwitz–Lerch zeta functions with applications. Appl Math Inf Sci 8:1485–1500

    MathSciNet  Google Scholar 

  • Srivastava HM (2019) The Zeta and related functions: recent developments. J Adv Eng Comput 3:329–354

    Google Scholar 

  • Srivastava HM, Aouf MK (1992) A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. J Math Anal Appl 171:1–13

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Aouf MK (1995a) A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. II. J Math Anal Appl 171:673–688

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Aouf MK (1995b) Some applications of fractional calculus operators to certain subclasses of prestarlike functions of negative coefficients. Comput Math Appl 30(1):53–61

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Attiya AA (2007) An integral operator associated with the Hurwitz–Lerch zeta function and differential subordination. Integral Transforms Spec Funct 18:207–216

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Bansal D (2017) Close-to-convexity of a certain family of \(q\)-Mittag–Leffler functions. J Nonlinear Var Anal 1(1):61–69

    MATH  Google Scholar 

  • Srivastava HM, Choi J (2012) Zeta and \(q\)-Zeta functions and associated series and integrals. Elsevier Science Publishers, Amsterdam

    MATH  Google Scholar 

  • Srivastava HM, El-Deeb SM (2020) A certain class of analytic functions of complex order with a q-analogue of integral operators. Miskolc Math Notes 21 (to appear)

  • Srivastava HM, Gaboury S (2015) A new class of analytic functions defined by means of a generalization of the Srivastava–Attiya operator. J Inequal Appl 2015:1–15

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Karlsson PW (1985) Multiple Gaussian hypergeometric series. Wiley, New York

    MATH  Google Scholar 

  • Srivastava HM, Owa S (eds) (1989) Univalent functions, fractional calculus, and their applications. Wiley, New York

    MATH  Google Scholar 

  • Srivastava HM, Owa S (eds) (1992) Current topics in analytic function theory. World Scientific Publishing Company, Singapore

    MATH  Google Scholar 

  • Srivastava R, Zayed HM (2019) Subclasses of analytic functions of complex order defined by \(q\)-derivative operator. Stud Univ Babeş Bolyai Math 64:69–78

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Yang D-G, Xu N-E (2009) Subordinations for multivalent analytic functions associated with the Dziok–Srivastava operator. Integral Transforms Spec Funct 20:581–606

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Răducanu D, Sălăgean GS (2013) A new class of generalized close-to-starlike functions defined by the Srivastava–Attiya operator. Acta Math Sinica (Engl Ser) 29:833–840

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Gaboury S, Ghanim F (2015) A unified class of analytic functions involving a generalization of the Srivastava–Attiya operator. Appl Math Comput 251:35–45

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Altınkaya S, Yalcin S (2018a) Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric \(q\)-derivative operator. Filomat 32:503–516

    MathSciNet  Google Scholar 

  • Srivastava HM, Gaboury S, Ghanim F (2018b) A study of meromorphically univalent functions defined by a linear operator associated with the \(\lambda \)-generalized Hurwitz-Lerch zeta function. Azerb J Math 8(1):35–51

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Juma ARS, Zayed HM (2018c) Univalence conditions for an integral operator defined by a generalization of the Srivastava-Attiya operator. Filomat 32:2101–2114

    MathSciNet  Google Scholar 

  • Srivastava HM, Prajapati A, Gochhayat P (2018d) Third-order differential subordination and differential superordination results for analytic functions involving the Srivastava-Attiya operator. Appl Math Inf Sci 12:469–481

    MathSciNet  Google Scholar 

  • Srivastava HM, Ahmad QZ, Khan N, Khan N, Khan B (2019a) Hankel and Toeplitz determinants for a subclass of \(q\)-starlike functions associated with a general conic domain. Mathematics 7:1–15

    Google Scholar 

  • Srivastava HM, Aouf MK, Mostafa AO (2019b) Some properties of analytic functions associated with fractional \(q\)-calculus operators. Miskolc Math Notes 20:1245–1260

    Google Scholar 

  • Srivastava HM, Khan B, Khan N, Ahmad QZ (2019c) Coefficient inequalities for \(q\)-starlike functions associated with the Janowski functions. Hokkaido Math J 48:407–425

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Mostafa AO, Aouf MK, Zayed HM (2019d) Basic and fractional \(q\)-calculus and associated Fekete-Szegö problem for \(p\)-valently \(q\)-starlike functions and \(p\)-valently \(q\)-convex functions of complex order. Miskolc Math Notes 20:489–509

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Raza N, AbuJarad ESA, Srivastava G, AbuJarad MH (2019e) Fekete-Szegö inequality for classes of \((p, q)\)-starlike and \((p, q)\)-convex functions. Rev Real Acad Cienc Exactas Fís Nat Ser A Mat (RACSAM) 113:3563–3584

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Tahir M, Khan B, Ahmad QZ, Khan N (2019f) Some general classes of \(q\)-starlike functions associated with the Janowski functions. Symmetry 11:1–14

    MATH  Google Scholar 

  • Srivastava HM, Tahir M, Khan B, Ahmad QZ, Khan N (2019g) Some general families of \(q\)-starlike functions associated with the Janowski functions. Filomat 33:2613–2626

    MathSciNet  MATH  Google Scholar 

  • Tang H, Khan Q, Arif M, Raza M, Srivastava G, Rehman SU, Ahmad B (2019) Some applications of a new integral operator in \(q\)-analog for multivalent functions. Mathematics 7:1–13

    Google Scholar 

  • Thilagavathi K (2017) Certain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator. Int J Pure Appl Math Sci 10:85–97

    Google Scholar 

  • Uçar HEÖ (2016) Coefficient inequality for \(q\)-starlike functions. Appl Math Comput 276:122–126

    MathSciNet  MATH  Google Scholar 

  • Wang Z-G, Raza M, Ayaz M, Arif M (2016) On certain multivalent functions involving the generalized Srivastava–Attiya operator. J Nonlinear Sci Appl 9:6067–6076

    MathSciNet  MATH  Google Scholar 

  • Wongsaijai B, Sukantamala N (2015) Applications of fractional \(q\)-calculus to certain subclass of analytic \(p\)-valent functions with negative coefficients. Abstr Appl Anal 2015:1–12

    MathSciNet  MATH  Google Scholar 

  • Xu N-E, Yang D-G, Sokół J (2013) A class of analytic functions involving in the Dziok–Srivastava operator. J Inequal Appl 2013:1–15

    MathSciNet  MATH  Google Scholar 

  • Xu Q-H, Xiao H-G, Srivastava HM (2014) Some applications of differential subordination and the Dziok–Srivastava convolution operator. Appl Math Comput 230:496–508

    MathSciNet  MATH  Google Scholar 

  • Yan C-M, Liu J-L (2019) Geometric properties of certain analytic functions associated with the Dziok–Srivastava operator. Symmetry 11(2):1–8

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    H. M. Srivastava

  2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, ROC

    H. M. Srivastava

  3. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    H. M. Srivastava

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Srivastava, H.M. Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis. Iran J Sci Technol Trans Sci 44, 327–344 (2020). https://doi.org/10.1007/s40995-019-00815-0

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