OUTLINE OF THE PROPOSED TOPIC OF

RESEARCH

Name of the Candidate : Gauri Shankar Paliwal

Roll No : 2012PhDENGG002

Enrollment No : 20120200232

Place of Research Work and Organization : JK Lakshmipat University, Jaipur

Proposed Supervisor Name : Dr. Jaya Gupta

Qualification : Ph.D.

Designation : Asst. Professor

Organization : JKLU, Jaipur

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CONTENTS

1. Proposed Topic of Research 32. Objective of the Proposed Research 33. Area of Proposed Research 34. Review of Literature 44.1. Classes of analytic functions 54.2. Fractional Operators 54.3. SUllie dcfiuitious and some staudanl classes of univalcut functions 94.4. Major subclasses ill literature defined using subordiuatiou 125. Research Gap 136. Methodology 147. Work Schedule 14References 14

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1. PROPOSED TOPIC OF RESEARCH

Classes of Analytic Functions Involving Fractional Integral Operators and itsApplications.

iit

2. OBJECTIVE OF THE PROPOSED RESEARCH

Objective of this study is to find gaps by review of literature and try to fill thesegaps by introducing some new classes of analytic functions defined in the open unitdisk and discuss their geometric properties.

(1) To define new class and find extreme points, redius of starlikeness, deriveFckcte-Szego type coefficient inequalities and its application with fractionaloperators by expressing it in convolution form.

(2) To study the inclusion property and other geometric properties by introduc-ing some new classes of analytic functions defined using Kornat u Integral.

(3) '1'0 discuss geometric properties by defining some new classes of analyticfunction using subordination and differential subordination.

(4) '1'0 obtain sufficient conditions of starlikeness, convexity and close-to-convexity,alpha-convexity, spiralikeness for the newly defined classes of analytic func-tions by using the principles of differential subordination.

(5) To find the applications of the above defined classes.

3. AREA OF PROPOSED RESEARCH

Geometric Function Theory and Fractional Calculus

Introduction:

Geometric Function Theory is that branch of complex analysis, which deals andstudies the geometric properties of the analytic functions. That is Geometric func-tion Theory is an area of Mathematics characterized by an intriguing marriagebetween geometry and analysis. Its origins date from the 19th century but newapplications arise continually. Interest in Geometric Function Theory has expe-rienced resurgence in recent decades as the methods of algebraic geometry andfunction theory on compact Riemann surfaces have found relevance in constructing'finite-gap' solutions to non-linear integrable system, a growing, specialized area ofMathematics with many connections to Mathematical Physics.

The geometric theory of complex variable functions was set as a separate branchof complex analysis in the 20-th century when appeared the first important papersin this domain, owed to P. Koebe [17], I.W. Alexander [3], L. Bieberbach [7].

The univalent function notion occupies a central role in geometric theory ofanalytic functions, first paper dating since 1907 owed to P. Koebe. The studyof univalent functions was continued by Plemelj, Gronwall and Faber. Even newdevelopments in the constructive approach to linear and non-linear boundary valueand initial value problems using spectral analysis [5] are likely to lead to a role forGeometric Function Theory in the solution of a wide range of partial differentialequations.

Geometric Function Theory is a classical subject. Yet it continue to find new ap-plications in an ever-growing variety of areas such as modern mathematical physics,

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more traditional fields of physics such as fluid dynamics, non-linear integrable sys-tems theory of partial differential equations. Geometric Function Theory is not asold as other branches of Mathematics. The first stirring of function theory is foundin the 18th century with Euler. Modern function theory was developed in the 19thcentury.

Fractional calculus is a field of mathematics study that grows out of the tra-ditional definitions of calculus integral and derivative operators in much the sameway fractional exponents is an outgrowth of exponents with integer value. The ba-sic mathematical ideas of fractional calculus (int.egral and differential operations ofnon integer order) were developed long ago by the mathematicians Leibniz (1695),Liouville (1834), Riemann (1892), and others and brought to the attention of theengineering world by Oliver Heaviside in the 18908, it was not until 1974 that thefirst book on the topic was published by Oldham and Spanier. Recent monographsand symposia proceedings have highlighted the application of fractional calculus inphysics, continuum mechanics, signal processing, and electromagnetics. The stud-ies have also found its use in studies of viscoelastic materials, as well as in manyfields of science and engineering including fluid flow, rheology, diffusive transport,clecterical networks, electromagnetic theory and probability.

4. REVIEW OF LITERATURE

The geometric theory of complex variable functions was set as a separate branchof complex analysis in the XX-th century when appeared the first important papersin this domain, owed to P. Koebe [17], I.W. Alexander [3], L. Bieberbach [7]. Theunivalent function notion occupies a central role in geometric theory of analyticfunctions, first paper dating since 1907 owed to P. Koebe. The study of univa-lent functions was continued by Plemelj, Gronwall and Faber. Now exist manytreated and monographs dedicated to univalent functions study. Among them werecall those of P. Mentel, Z. Nehari, L.V. Ahlfors [1], Ch. Pommcrcnkc [34], A.W.Goodman [11], P.L. Duren, D.J. Hallenbeck, T.H. MacGregor [13], S.S. Miller, P.T.Mocanu [23] and P.T.Mocanu, T. Bulboacoa, Gr. St. Salagean [24].

The theory of univalent functions is one of the most beautiful subjects in Geo-metric Function Theory. Its origin (apart from the Riemann mapping Theorem)can be traced to the 1907 paper of Koebe [17], to Gronwall's proof of the AreaTheorem in 1914 and to Bieberbach's estimate for the second coefficient of normal-ized univalent functions in 1916 and its consequences. By then, univalent functiontheory was a subject in its own right. The Mathematical Romanian School broughther valuable contribution in the geometric theory of univalent functions. Amongthem we mention two personalities from Cluj, namely G.C Aalugareanu and P. T.Mocanu. G. C Alugareanu, the creator of the romanian school's theory of univalentfunctions, was the first mathematician who obtain in 1931 the necessary and suffi-cient conditions for univalence in the open unit disc. The researches initiated by G.C. Alugareanu are continued by P. T. Mocanu, who obtained important results inthe geometric theory of univalent functions: introducing alpha-convexity, gettingunivalence criteria for non analytic functions, development in collaboration with S.S. Miller the method of differential subordinations, and 1f1OStrecently the theoryof differential superordiuations, The method of differential subordinations has auimportant role in a much easier demonstration of already known results, as well asin many other new obtained results.

...

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4.1. Classes of analytic functions. The class S* of starlike function was firststudied by Alexander [3]. In 1936 Rebertson [36] introduce the class S* (p) of starlikefunction of order p. In 1936 .Robertson [36] introduced the class C(p) of convexfunction of order p , 0 ::; p < 1. In 1969, P.Mocannu [25] introduced the conceptof alpha- convex and alpha - starlike functions. In 1973, [22] it was proved that alla-convex function are convex if a ;:::1 and starlike if a < 1. The class of close - to -convex function introduced by Kaplan [15] in 1952 and denoted by K. The class Khas been introduced independently and in a quite different manner by Biernacki [8].The close-to-convex functions are univalent, (see [15]) and the class K proved the

!!J!! most useful subclass of S .We note that convex and starshaped domains are close-to-convex. The class K is of considerable importance in as much as it containsmost of the known subclasses of S, this can be summarized as: C c S' eKe S.

Noor [30] Considered a new subclass C* of univalent function. The functions inC' is called quasi-convex functions and C c C' eKe S. It is shown (see [30]) f EC' iff z ]' E K. The function f called a- quasi -convex function has been defined andits properties were studied in [29]. The class of a- quasi -convex functions is denotedby Qu. In 1970 , Robertson [37], introduced the concept of Quasi Subordination.In 1934/35 Noshiro and Warchawaski obtained a simple but interesting criterionfor univalence of analytic functions. They proved that if an analytic function fsatisfies Re{f'(z)} > 0 for all z E E, then f is close-to-convex and hence univalentin E.

AI-Amiri and Reade [2], in 1975, have shown that for a ::;0 and also for a = 1,the functions f E A, satisfying the differential inequality R[I(a, f(z))] > 0; z E E,are univalent in E. In 2005, Singh, Singh and Gupta [46] proved that for 0 < a < 1,functions f E A, satisfying the differential inequality R[I(a,f(z))] > a;z E E, areunivalent in E. The univalence of the above problem is still open for a > 1.

Janowski introduced the class P[A, B], for A and B. Classes SOlA, B] andK[A, B] are introduced by K. S. Padmanabhan [32]. Ma and Minda [21] intro-duced the classes S*(¢) and C(¢). The class S*(¢) reduce to the class S*[A,B]of Janowski Starlike function. For 0 < a ::;1, Parvatham introduced and studiedthe S*[a]. Following Ma and Minda [21], the authors defined a more general classof starlike functions of complex order s; (¢) [35] and convex functions of complexorder Cb(¢) [35].

R( a, (3) is the class of functions a-Prcstarlike of order /3 and was introduced bySheil- Small, Silverman and Silva [44]. C(a, (3) is called the class of functions of a-preconvex of order /3 .

Definition 4.1.1. If f (z) and g(z) are given by f(z) = z+ L::~=2ar.z" and g(z) =z + L::~=2b.,':" then Conooiution or Hadamard Product of f(z) and g(z) is

00

h(z) = (J * g)(z) = z + L a,.bnzn

n=2

Definition 4.1.2. Faltung Product(J * *g) defines as

00 o-b-. nJ(z) = f(z) * *g(z) = z + L-z

n=2 n

4.2. Fractional Operators.

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Definition 4.2.1. Let f(z) be analytic in a simply connected region of the z-planecontaining the origin. The fractional derivative of f of order ,\ is defined by

AId r f(OD, f(z) := ['(1 _ .\) dz Jo (z _ 0'\ d(,

where the multiplicity of (z - <V is removed by requiring that log(z - 0 is real for(z - 0> 0Definition 4.2.2. Fractional Derivative of order (n + ,\) [49] is defined as:

d'1I.(D~H).f(z) = d.:" D;f(·:) (0:::;'\ < 1:n E Nu)

Definition 4.2.3. Let f(z) is all analytic function in a. simply connected region ofthe z-planc containing the origin, and the multiplicity of (z - 0" is removed byrequiring that log( z - () to be real when (z - () > O. Then the generalized fractionalderivative of order .\ is defined for a. function f(z) by (see, e.g. [48])(4.2.1)

{

r(l ~.\):z {ZA-,1 fa' (z - O-A.2F1 (11-'\, -l/; 1-.\; 1- n f(()d( LJo~:1/ f(z) = (0:::; .\ < 1)

dn-d Jt~n,IJ.,1/ f(z), (n :::;,\ < n + 1, n E N)zn ,

and f(z) = O(lzlk), (z -+ 0, k > max{O, 11- u - I} - 1)It follows at once from the above definition that

Furthermore, in terms of gamma function, we have:

]'\111/ f' _ 1'(p+1)r(p-Il+//+1) Zf'-11

, 0 Z Z .- r(p - 11+ 1)r(p - .\ + l/ + 2)

(0:::; .\; {J> ma:i{O·Il- l/ - I} - 1)Definition 4.2.4. Fractional Integral of order .\ [49] is defined [or a [unction r, by

D-;'\f(z) = rt.\) fa' (z !~(/l-Ad((.\ < 0)

Where f(z) is an analytic functions in a simply connected region of the complexz-plane containing the origin (z=O) , and the multiplicity of (z - 0,\-1 is removedby requiring log (z - Oto be real when z - ( > o. -Definition 4.2.5. The operator Ig,'f'ry is a generalization of the fractional integraloperator Ig;f·T/ introduced by Saigo [39]. For real numbers a > 0, f3 and T), thefractional i~tegra.l operator defined by

z-o-/3 r ( ()Ig:!"/f(z) = r(a) Jo (z - (),-,-12Fl a + f3, -TJ; a; 1- -; f(()d(

Where f(z) is an analytic function in a simply connected region of the z- planecontaining the origin, with the order f(z) = O(lzlf)(Z -+ 0) where E > max{O,f3-TJ} - 1 and the multiplicity of(z - 0,,-1 is removed by requiring log(z - 0 to byreal when (z - () > (I.

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r:

Definition 4.2.6. The fractional integral operator J~':'" [16] is introduced bySrivastava and defined as:

Let 0 > 0, min(o + TJ, -/3 + TJ, -/3) > -2 and 3 ~ (3(G,t'l) then the fractionalintegral operator defined by

.1G.{3 .n f(z) = r(2 - /3)r(2 + 0 + TJ) zO IG.{3 •." f(z)0,. r(2 - /3 + TJ) 0,.

Definition 4.2.7. Ruscheweyh Derivatives of (n + p - l)th order [26] denoted byDn+p-l : A(p) -t A(p) and defined by

zPtr+v:' f(z) = * f(z) (f(z) E A(P»(1 - z)n+P

zp(zn-l f(z»n+p-l(n+p-l)!

The symbol Dn+p-l when p =1 was introduced by Ruscheweyh [38],and the symbolDn+p-l was introduced by Goel and Sohi [10].

recurrence relation [26]:Z(Dn+1J

-1f(z))' = (n+p)D"+IJf(z) - nD"+P-lf(z)

Definition 4.2.8. Generalized Ruscheweyh derivative introduced by Goyal andGoyal [33] is defined as

J>'.I-' f( ) := r(J..l- A + 1/ + 2) .1>'1-'.//( 1-'-1 f(z»z I'(J..l + 1)1'(1/ + 2) z 0,. z

00

= z + Lan B>"I-' (n)znn=2

whereB)"~'(n)'- r(n + J..l)r(1/+ 2 + J..l - A)r(n + 1/ + 1)

.- r(n)r(n + 1/ + 1 + J..l- A)r(1/ + 2)r(1 + J..l)For J..l = A , generalized Ruscheweyh derivative J>',/1' reduces to the ordinary Ruscheweyhderivative D). of order A .

Convolution:

Jo\~"vzP =2 F1(J..l + 1,1/ + 2; 1/+ 2 + J..l- A; z) * f(z)

Definition 4.2.9. Let the function f E A, form,n E No = NU{O}, 0 ~ Al ~ A2,E. A. Eljamal and M. Darus [9] define the operator

D';": f(z) = Z + ~ [1 + (AI + A2)(k - 1)]rrt C(n k)a.zk>'I.>'~ 0 1+A2(k-l) , k

k=H+I

Which is generalized Ruscheweyh derivatives operator.Definition 4.2.10. For f(z) E A, p analoque of Salagean Operator introduced byShenen [45]which is the extension of Salagean Operator defined by Salagean [40]

D~·Pf(z)=zP+ f [l+(~-l)A]\kZk, zEEk=n+p p

D~'Pf(z) = f(z)A

D~'P f(z) = (1 - A)f(z) + -zf'(z) = D,d(z)p

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D~'Pf(z) = DA(D~-1'Pf(z))For>. = 1, p=l this p analogue reduce to be Salagean Operator introduced bySalagean [40]

Note that,co

Dnf(z) = z +L knakzk, n E No = {o} UNk=2

Definition 4.2.11. Owa and Srivastava [31] introduced the operator n" : A -+ Adefined by

nA f(z) = 1'(2 - >')ZAD;f(z) (/\ =I- 2,3,4 .... )

The operator 12" is called Oioa -Srioastaoa Operata'!' [49].Definition 4.2.12. AL- Oboudi Differential Operator [4] defines 8S:

DOf(z) = f(z)D1 f(z) = (1 - ,\)f(z) + /\::/,(z) = D.\f(z).Dnf(z) = D,,(Dn-l f(z))

= z + 2:~2[1 + (k - Ip]nakzkIf we put>. = 1, we have Salagean operator.

Definition 4.2.13. Let n E N U {O} and), 2: 0. By n~we denote the Najafzadehoperator [28] n~:N -+ N defined by

n~f(z) = (1 - )')S" f(z) + )'Rn f(z)Salagean Operator [40]Ruscheweyh differential operator [38]

S"] -+R''] -+

For univalent function

nrjJ(z) = z - f [kn(1 _),) +>. (k +: - 1)] akzkk=+1

Also n~f(z) = f(z)Definition 4.2.14. Schawarzian Derivative of w with regards to z [42]

(") 1 ( ")2 __ Will __ 3 (WII)2(/3(W) = {w.z} = ~ - - ~

ui' 2 ui' Wi 2 u)'

(4.2.2)

And

ieWilli Willi ui" (Wll ) 3

(/4(W) = - - 6-- +6 -Wi W/2 tu'

Definition 4.2.15. For function f(z) E A , Libra integral operator defined as

21'L{f(z)} = - f(t)dtz 0

This operator introduced by Libra [19].Definition 4.2.16. For a function f(z) belonging to the class A, generalized Libraintegral operator defined as:

Lc{f(z)} = c;; 1l' tc-1 f(t)dt (C 2: 0)

The operator t.; when C EN was introduced by Bernardi [6].

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Recurrence relation:

(1 + c)f(z) = cL,f(z) + z(Lcf(z))'Definition 4.2.17. Bernardi Operator (J1 .• ) [27)defined as,

+ 11'(Jj.I.)f(z) = _J.l_ tl.1-1f(t)dt == Hi(J.l + 1, 1;J.l+ 2)f(z)zj.l. 0

Note that the operator was studied earlier by Libra [20J.Definition 4.2.18. Komatu Integml Operator [41], [18Jis defined as

K6 f(z) = (c + p)6 r tC-l(log~)6-1f(t)dtc.p r(c5)zc Jo t

For p-valent function

K~,p f (z) = zp +f (c: ; : k) a ap+kzP+kk=l

Definition 4.2.19. Genemlization of Komatu integml operator has been given byT. O. Salim [41)directly for p-valent functions

J~:~6Af(z) = zp+ ~ [C~;:kr (1 + ~:)]ffiap+kzp+K

For m=l.'\ = 0 it reduces to Komatu Integral Operator.J~.~.Af(z) = f(z)J;.:.Af(z) = (1 - ,\)KZ,pf(z) + AI"(KZ,pf(z)) = Jg.p.Af(z)J~~6Af(z) = Jg.p.A(J';p~i.·A f(z))Remarks [41):J;'110 == i; Barnardi-Iibra-Livingston operator [6)J;'jOA == Dr;: Generalized Salagean operator [4)J:f'i°o == o» Salagean operator [40)R~~urrence Relation [41) :

z{J;,'~~Af(z)} = (c + P)f:;~A-* f(z) - cJ;,'~\f(z)

4.3. Some definitions and some standard classes of univalent functions.Definition 4.3.1. A complex function is said to be analytic function on a region Rif it is complex differentiable at every point ill R. The terms holomorphic function,differentiable function, and complex differcutiable function are sometimes used in-terchangeably with analytic function. A denote the class of functions f (z) of theform

oc

(4.3.1) f(z) = z + Lanz"

which are analytic in the open unit disk E = {z : z E Candlzl < I}.Definition 4.3.2. A function f analytic in a domain D is said to be univalentfunction there if it does not take the same value twice, that is f(zd =1= f(Z2)for all pairs of distinct points Zl and Z2 in D. In other words f is one-to-one (orinjective) mapping of D onto another domain. S be the subclass of A consisting ofall univalent functions in E.

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Definition 4.3.3. The class P is the set of all functions of the form00

P() 1 2 3 n 1 '" nZ = + C1Z + C2Z + C3Z + ...+ CnZ ... = + L CnZ

n=l

that is analytic in E and such that for Z E E ,

. . 1+ ZRe{p(z)} > 0 ie p(z) E P Iff p(z) -< 1 _ z

Any function in P is called a functions with positive real part in E.It should be noted that p(z) is not necessarily required to be univalent. Thus

p(z) = 1 + zTL is in P for any integer n ~ 0 , but if n ~ 2 the function is notunivalent.Definition 4.3.4. The set E is said to be Starlike with respect to a point Wo if thelinear segment joining any point of E and Wo lies entirely in E. The class S' wasfirst studied by Alexander [3] let: f E A , f(O) = 0 and f if; univalent in E , Then fmaps E onto a starlike domain w. r. to Wo iff

[;;/,(z)]

Re 7(Z) > 0, z E E

Definition 4.3.5. In 1936 Robertson [36] introduce the class S*(a). If f E A issaid to be in the class S* (a) of univalent starlike functions 01 order a if

[ZII(z)]Re l(z) > a, (0 S a < 1, z E E)

We write S' (0) = S' ,the class of univalent starlike in E.Definition 4.3.6. The set E is said to be convex if it is starlike with respect toeach of its points; that is, if the linear segment joining any two points of E liesentirely in E. A convex function is one which maps the unit disk conform ally ontoa convex domain. The class of all convex functions denoted by C let 1E A and let1be univalent in E . Then 1maps E onto a convex domain iff

[z1"(z)]

Re 1+ 1'(z) >0, zEE

This condit.ion was first stated and Study by Sheil-smail [47J.

Definition 4.3.7. III 1936, Robertson [36] introduced the class C(p) of convexfunction of order p, 0 S p < 1 which are defined by

{ ( z1"(z)) }C(p)= lEA:Re 1+ 1'(z) >p, OSp<l, zEE

In particular C(O) = C, Thus C c S' c SDefinition 4.3.8. A function 1E A is said to be close- to- convex iff

Re [z/,(z)] > O,z E Eg(z)

for some 9 E S' or Equivalently if

[ZII(z)]Re GI(Z) > 0, z E E for some G E C

This class introduced by Kaplan in 1952 [15] and denoted by K.

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The class K has been introduced independently and in a quite different mannerby Biernacki [8]. The close-to-convex functions are univalent, see [15] and theclass K proved the most useful subclass of S .We note that convex and starshapeddomains are close-to-convex. The class K is of considerable importance in as muchas it contains most of the known subclasses of S, this can be summarized as:

C c S* c tc c S

Definition 4.3.9. In 1969, Mocannu [25] introduced the concept of alpha- convexand alpha - starlike [unctions. A function 1 is said to be a - convex in the openunit disk E if it is analytic,

l(z)/'(z) ~o and Re[(1_a)zl'(z) + a (zf'(z))'] >0 zE Ez f(z) f'(z) ,

The set of all such function is denoted by MoIf a = 1 , then a-convex function is convex and if a = ° then a-convex function

is starlike.In 1973 [22], it was proved that all a-convex function are convex if a ~ 1 and

starlike if a < l.

Definition 4.3.10. Noor [30] considered a new subclass C' of univalent functionie f E A belongs to C' iff f is analytic in E and is such that there exists z E Csatisfying

Re [(z1'(z))'] > o. z E Eg'(2)

The function in C' is called quasi-convex and

C c C' c tc c S

Note: 1 E C' if),z1'(z) E K .(see [30] )Definition 4.3.11. The function 1 called a-quasi convex [unction. has been definedand its properties studied in [29]. A function f, analytic in E, is said to be a- quasiconvex iff there exists 9 E C such that, for a real and positive

Re [(1 - a) f'(z) + a (z1'(z))'] > 0, z E Eg'(z) g'(z)

The Class of a- quasi convex function is denoted by QCtDefinition 4.3.12. For functions 1and g, analytic in 6, we say that 1 is subordi-nate to 9 if there exists a Schwarz function w(z) analytic in 6, with w(o) = ° andIw(z)1 < Izl < 1 (z E 6) Such that

f(z) = g(w(z)) (z E 6)

We denote this subordination as f -< 9 or f(z) -< g(z) (z E 6) (see [12])Definition 4.3.13. A sequence{b.,}k=1 of complex number is said to be a subor-dinatiru; factor sequence(see [12]) if for the functionf(z) E A is analytic, univalentand convex in U. We have the subordination given by

00

L akbkzk -< I(z) (z E E; al = 1)k=l

iii

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Definition 4.3.14. In 1970 , Robertson [37]introduced the concept of Quasi Subor-dination. For two analytic functions f and 9 , the function f is Quasi Subordinationto 9 written as

fez) -<q g(z)If there exists analytic functions </Jand w, with 1</J(z)1 ::; 1, w(O) = 0 and Iw(z)1 <

1 such thatfez) = </J(z)g(w(z))

observe that </J(z)= 1 then fez) = g(w(z)) so that fez) -< g(z)Note:Also notice that if w(z) = (z) then fez) = </J(z)g(z) and it is said that f is

Majorised by 9 and written as

fez) «g(z) inE

4.4. Major subclasses in literature defined using subordination.Definition 4.4.1. Janowski Class F[A, B)(sce [14],):

Janowski introduced the class P[A, B] . for A and B ,-1 ::; B < A ::; 1, afunction p analytic in E with p(O)= 1 belongs to the class P[A, B] if

1 + Az 1 + Azp(z) is subordinate to --B ie p(z) -< --B-

1+ z 1+ zIn particular P[A, B] C P[l, -1] = PWe also note that

1- AB A- Bp E P[A, B] iff Ip(z) - 1 _ B2 I < 1 _ B2' (B =I ±1, z E E)

In addition it is known that P[A, B] is a convex set.Furthermore, it can easilybe shown that p E P[A, B] iff there exists a function h E P such that

( )_ (1 - A) + (1 + A)h(z)

p z - (1 _ B) + (1 + B)h(z)

Special selection of A and B lead to a familiar sets defined by inequalities. Some ofthese are given below under condition p(O) = 1 and 0 ::; p < 1

(i) P[l, -1] is the class P of function p with positive real part(ii)P[l - 2p, -1] is the class pep) of function p with Re{p(z)} > p in E(iii) P[l,O] is the class of function defined by Ip(z) - 11 < 1

Definition 4.4.2. Classes Introduced By Ma And Minda:Ma and Minda [21]introduced the classes S* (<p) and C(</J)by

* { zf'(z) }S (</J)= f E A: fez) -< </J(z)

When

</J(z)=~~~:, (-l::;B::; A::; 1)

The class S* (</J)reduce to the class S* [A, B] of Janowski Starlike function.Thus S' [A B] = S* (ItA,), ItB.Also S*[l, -1] = S' O~~)= S*For 0::; 0 < 1, the class S*[l - 20, -1] is the class S*(o) of starlike function

of order 0 .

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An equivalent analytic description of S* (0) is given by

S'(a)={fEA;Re(z;~;~)) >0, o~a<l}

for 0 ~ Q < 1

S*(a) = S*[I- 0,0] = {f E A: IZ;~;~)-11< 1- a}For 0 < a ~ 1 Parvathom introduced and studied the S* [0]Where

•

S*[ ] = S*[ - ,] = {f A' zf'(z) 1+a(z)}a 0, a E. f(z) -< 1 _ a(z)

= {f E A : Iz;~;~)- 11 < a Iz;~;~)+ 11 ,z E E,O < a ~ 1 }(For Convex)Ma and Minda introduced the c1assC(¢) by

C(¢) = {f E A: 1+ z;:~~~)-< ¢(z)}

When

¢(z)=~:~: (-I~B~A~I)

Class C(¢) reduce to the class C[A. B] of Janowski Convex function,

thus C[A. B] = C (:!~:)Also C = C[I, -1] = C (::~)For 0 ~ a < 1 ,C(a) = C[1 - 20, -1] is this class of convex functions of order a

C( ,) = {f A' 1 zf"(z) 1 + (1 - 2a)z}Q E. + f'(z) -< 1 - z

= {f E A : Re(1 + z;:~~~))> 0, 0 ~ a < 1}

The transform Joz ¥dt is called the Alexander transform of f(z).It is clear that f E C(a) iff zf' E S; or equivalently f E S; iff Alexander

_ transform of f(z) is in C(a) (see [43])

5. RESEARCH GAP

It has been observed that the authors have been defining new classes ami sub-classes of analytic functions but there is a need to continue on developing the newclasses and to investigate into the properties. however. continuous literature reviewis needed to support the work as the proposed work is dynamic in nature. Thisobservation is iufiueuced by its applications ill the field of science ami engiueeriugwhich require supported mathematical tools for its increasing requirements. A vastscope is also found to highlight the applications of this work .

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14

6. METHODOLOGY

The following methods are proposed to be used to undertake the proposed work:1. Differential subordinations2. MethoJ of differential inequalities3. Methods arising from the convolution theory.These methods are also proposed to be used to find new theorems and to discuss

the geometric properties of the defined classes.Mathematical softwares like Mathematica and Matlab are proposed to be used

to visualize these geometric properties.

7. VVORK SCHEDULE

We are expecting to finish the work by 30 mouths from the date of commence-ment. Following is the tentative plan of action:Phase 1: Introduction and review of literature

We are expecting to finish introduction and review of literature ill (U-12)lllUlltht;.Continuous literature survey will be carried out to collect the information requiredduring various stages of thc proposed research till thesis writing.Phase 2: Study of geometric properties of classes of analytic function

In this phase, we will find the gaps by review uf literature in phase 1 and fill thesegaps by introducing some classes of analytic function and study their geometricproperties and finding S01l1enew results.

We are expecting to finish this phase in (o-24)months.Phase 3: Conclusion and thesis writing

In this phase, all the work carried out in different phases will be documented inthe form of thesis.

We are expocting to finish this phase in (24-30)months.

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