Skip to main content

Classes of multivalent analytic and meromorphic functions with two fixed points

Abstract

The object of the present paper is to investigate the coefficients estimates, distortion properties, the radii of starlikeness and convexity, subordination theorems, partial sums and integral mean inequalities for classes of functions with two fixed points. Some remarks depicting consequences of the main results are also mentioned.

MSC:30C45, 30C50, 30C55.

1 Introduction

Let denote the class of functions which are holomorphic in D=D(1), where

D(r)= { z C : 0 < | z | < r } .

By M(p,k), where p, k are integer, p<k, we denote the class of functions fM of the form

f(z)= a p z p + n = k a n z n (zD; a p >0).
(1)

We note that for p<0 we have the class of functions which are meromorphic in U:= U 1 , U r := D r {0}, and for p0 we obtain the class of functions which are analytic in U.

Let p>0, α0,p), r(0,1. A function fM(p,k) is said to be convex of order α in D(r) if

( 1 + z f ( z ) f ( z ) ) >α ( z D ( r ) ( f ) 1 ( { 0 } ) ) .

A function fM(p,k) is said to be starlike of order α in D(r) if

( z f ( z ) f ( z ) ) >α ( z D ( r ) f 1 ( { 0 } ) ) .
(2)

We denote by S p c (α) the class of all functions fM(p,p+1), which are convex of order α in D and by S p (α) we denote the class of all functions fM(p,p+1), which are starlike of order α in D.

Let BM(p,k), p>0. We define the radius of starlikeness of order α and the radius of convexity of order α for the class by

R α ( B ) : = inf f B ( sup { r ( 0 , 1 ] : f  is starlike of order  α  in  D ( r ) } ) , R α c ( B ) : = inf f B ( sup { r ( 0 , 1 ] : f  is convex of order  α  in  D ( r ) } ) ,

respectively.

We say that a function f:UC is subordinate to a function F:UC, and write f(z)F(z) (or simply fF), if there exists a function ωM (ω(0)=0, |ω(z)|<1, zU), such that

f(z)=F ( ω ( z ) ) (zU).

In particular, if F is univalent in U, we have the following equivalence:

f(z)F(z) [ f ( 0 ) = F ( 0 ) f ( U ) F ( U ) ] .

For functions f,gM of the form

f(z)= n = 0 a n z n ,g(z)= n = 0 b n z n (zD),

by fg we denote the Hadamard product (or convolution) of f and g, defined by

(fg)(z)= n = 0 a n b n z n (zD).

For multivalent function fM(p,k), the normalization

z 1 p f(z) | z = 0 =0and z p f(z) | z = 0 =1
(3)

is classical. One can obtain interesting results by applying Montel’s normalization (cf. [1]) of the form

z 1 p f(z) | z = 0 =0and z p f(z) | z = ρ =1 ( ρ = | ρ | e i η ) ,
(4)

where ρ is a fixed point from the unit disk U. We see that for ρ=0 the normalization (4) is the classical normalization (3).

Let us denote by M ρ (p,k) the class of functions fM(p,k) with Montel’s normalization (4). It will be called the class of functions with two fixed points.

Also, by T η (p,k), ηR, we denote the class of functions fM(p,k) of the form

f(z)= a p z p n = k | a n | e ( n + p ) η z n (zD).
(5)

In particular, we obtain the class T 0 (p,k) of functions with negative coefficients. Moreover, we define

T(p,k):= η R T η (p,k).
(6)

The classes T(p,k) and T η (p,k) are called the classes of functions with varying argument of coefficients. The class T(1,2) was introduced by Silverman [2] (see also [3]). It is easy to show that for fT(p,k), p>0, the condition (2) is equivalent to the following:

| z f ( z ) f ( z ) p|<pα ( z D ( r ) f 1 ( { 0 } ) ) .
(7)

Let A, B, δ be real parameters, δ0, 0B1, 1A<B, and let φ,ϕM(p,k).

By W=W(p,k;ϕ,φ;A,B;δ) we denote the class of functions fM(p,k) such that

(φf)(z)0(zD)
(8)

and

( ϕ f ) ( z ) ( φ f ) ( z ) δ| ( ϕ f ) ( z ) ( φ f ) ( z ) 1| 1 + A z 1 + B z .
(9)

If 0<B<1, then the function

h(z)= 1 + A z 1 + B z (zD)
(10)

is univalent in U and maps U onto the disk {wC:|wa|<R}, where

a= 1 A B 1 B 2 ,R= B A 1 B 2 .

Thus, by definition of subordination the condition (9) is equivalent to the following:

| ( ϕ f ) ( z ) ( φ f ) ( z ) δ| ( ϕ f ) ( z ) ( φ f ) ( z ) 1| 1 A B 1 B 2 |< B A 1 B 2 (zD).
(11)

If B=1, then the function (10) maps the disc D onto the half-plane {wC:[w]> 1 + A 2 }. Thus, the condition (9) is equivalent to the following:

δ| ( ϕ f ) ( z ) ( φ f ) ( z ) 1| { ( ϕ f ) ( z ) ( φ f ) ( z ) 1 } < 1 A 2 (zD).
(12)

Now, we define the classes of functions with varying argument of coefficients related to the class W=W(p,k;ϕ,φ;A,B;δ). Let us denote

W ρ = W ρ ( p , k ; ϕ , φ ; A , B ; δ ) : = A ρ ( p , k ) W ( p , k ; ϕ , φ ; A , B ; δ ) , T W η = T W η ( p , k ; ϕ , φ ; A , B ; δ ) : = T η ( p , k ) W ( p , k ; ϕ , φ ; A , B ; δ ) , T W ρ η = T W ρ η ( p , k ; ϕ , φ ; A , B ; δ ) : = M ρ ( p , k ) T W η ( p , k ; ϕ , φ ; A , B ; δ ) , T W ρ = T W ρ ( p , k ; ϕ , φ ; A , B ; δ ) : = T ( p , k ) W ρ ( p , k ; ϕ , φ ; A , B ; δ ) .

The class W=W(p,k;ϕ,φ;A,B;δ) unifies various new and also well-known classes of analytic or meromorphic functions; see for example [136].

For the presented investigations we assume that φ, ϕ are the functions of the form

φ ( z ) = z p + n = k α n z n , ϕ ( z ) = z p + n = k β n z n ( z D ) , 0 α n < β n ( n N k : = { k , k + 1 , } ) .
(13)

Moreover, let us put

d n :=(δ+1)(1+B) β n ( δ ( B + 1 ) + A + 1 ) α n (n N k ).
(14)

The object of the present paper is to investigate the coefficients estimates, distortion properties, the radii of starlikeness and convexity, subordination theorems, partial sums and integral mean inequalities for the classes of functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.

2 Coefficients estimates

We first mention a sufficient condition for the function to belong to the class W.

Theorem 1 Let 0B1 and 1A<B. If f M ρ (p,k) and

n = k d n | a n |(BA) a p ,
(15)

then fW.

Proof If 0B<1, then we have

| ( ϕ f ) ( z ) ( φ f ) ( z ) δ | ( ϕ f ) ( z ) ( φ f ) ( z ) 1 | 1 A B 1 B 2 | ( δ + 1 ) | ( ϕ f ) ( z ) ( φ f ) ( z ) 1 | + B ( B A ) 1 B 2 ( δ + 1 ) n = k ( β n α n ) | a n | | z | n p a p n = k α n | a n | | z | n p + B ( B A ) 1 B 2 .

Thus, by (15), we obtain (11) and consequently fW. Let now B=1. Then simply calculations give

δ | ( ϕ f ) ( z ) ( φ f ) ( z ) 1 | { ( ϕ f ) ( z ) ( φ f ) ( z ) 1 } ( δ + 1 ) | ( ϕ f ) ( z ) ( φ f ) ( z ) 1 | ( δ + 1 ) n = k ( β n α n ) | a n | | z | n p a p n = k α n | a n | | z | n p .

Thus, by (15) we obtain (12). Hence fW and the proof is complete. □

Theorem 2 Let f T η (p,k). Then f TW η if and only if the condition (15) holds true.

Proof Let f TW η . In view of Theorem 1, we need only show that f satisfies the coefficient inequality (15). Putting z=r e i η in the conditions (11) and (12) we obtain

(δ+1) n = 2 ( β n α n ) | a n | r n p a p n = 2 α n | a n | r n p < B A 1 + B .

By (8), it is clear that the denominator of the left hand side cannot vanish for r0,1). Moreover, it is positive for r=0, and in consequence for r0,1). Thus, we have

n = 2 d n | a n | r n p <(BA) a p ,

which, upon letting r 1 , readily yields the assertion (15). □

By applying Theorem 2, we can deduce following result.

Theorem 3 Let f T η (p,k). Then f TW ρ η if and only if it satisfies (4) and

n = k ( d n ( B A ) | ρ | n p ) | a n |BA.
(16)

Proof For a function f T η (p,k) with the normalization (4), we have

a p =1+ n = k | a n ||ρ | n p .
(17)

Then the conditions (15) and (16) are equivalent. □

From Theorem 3, we obtain the following lemma.

Lemma 1 Let there exist an integer n 0 N k such that

d n 0 (BA)|ρ | n 0 p 0.
(18)

Then the function

f n 0 (z)= ( 1 + a ρ n 0 p ) z p a e i ( p n 0 ) η z n 0

belongs to the class TW ρ η for all positive real numbers a. Moreover, for all n N k such that

d n (BA)|ρ | n p >0,
(19)

the functions

f n (z)= ( 1 + a ρ n 0 p + b z n p ) z p a e i ( p n 0 ) η z n 0 b e i ( p n ) η z n ,

belongs to the class TW ρ η for all positive real numbers a and

b= B A + ( ( B A ) | ρ | n 0 p d n 0 ) a d n ( B A ) | ρ | n p .

By Lemma 1 and Theorem 3, we have following two corollaries.

Corollary 1 Let

d n (BA)|ρ | n p 0(n N k ).

If

d n (BA)|ρ | n p >0,

then the nth coefficient of the class TW ρ η satisfies the following inequality:

| a n | B A d n ( B A ) | ρ | n p .
(20)

The estimation (20) is sharp, the function f n , η of the form

f n , η (z)= d n z p ( B A ) e i ( p n ) η z n d n ( B A ) | ρ | n p (zD)
(21)

is the extremal function.

Corollary 2 If

d n (BA)|ρ | n p =0,

then the nth coefficient of the class TW ρ η is unbounded. Moreover, if there exists n 0 N k such that

d n 0 (BA)|ρ | n 0 p <0,

then all of the coefficients of the class TW ρ η are unbounded.

By putting ρ=0 in Theorem 3 and Corollary 1, we have the corollaries listed below.

Corollary 3 Let f T η (p,k). Then f TW 0 η if and only if

n = k d n | a n |BA.
(22)

Corollary 4 If f TW 0 η , then

a n B A d n (n N k ).
(23)

The result is sharp. The functions f n , η of the form

f n , η (z)= z p B A d n e i ( p n ) η z n (zD;n N k )
(24)

are the extremal functions.

3 Distortion theorems

From Theorem 2, we have the following lemma.

Lemma 2 Let f TW ρ η . If the sequence { d n } satisfies the inequality

0< d k (BA)|ρ | k p d n (BA)|ρ | n p (n N k ),
(25)

then

n = k | a n | B A d k ( B A ) | ρ | k p .

Moreover, if

0< d k ( B A ) | ρ | k p k d n ( B A ) | ρ | n p n (n N k ),
(26)

then

n = k n| a n | k ( B A ) d k ( B A ) | ρ | k p .

The second part of Lemma 2 may be formulated in terms of σ-neighborhood N σ defined by

N σ = { f ( z ) = a p z p + n = k a n z n T η ( p , k ) : n = k n | a n | σ }

as the following corollary.

Corollary 5 If the sequence { d n } satisfies (26), then T W ρ η N σ , where

δ= k ( B A ) d k ( B A ) | ρ | k p .

Theorem 4 Let f TW ρ η , |z|=r<1. If the sequence { d n } satisfies (25), then

ϕ(r)|f(z)| d k r p + ( B A ) r k d k ( B A ) | ρ | k p ,
(27)

where

ϕ(r):={ r p ( r ρ ) , d k r p ( B A ) r k d k ( B A ) | ρ | k p ( r > ρ ) .
(28)

Moreover, if (26) holds, then

p a p r p 1 k ( B A ) d k ( B A ) | ρ | k p r k 1 | f (z)| p d k r p + k ( B A ) r k 1 d k ( B A ) | ρ | k p .
(29)

The result is sharp, with the extremal function f k , η of the form (21) and f 0 (z)=z.

Proof Suppose that the function f of the form (1) belongs to the class TW ρ η . By Lemma 2 we have

| f ( z ) | = | a p z p + n = k a n z n | r p ( a p + n = k | a n | r n p ) r p ( 1 + n = k | a n | | ρ | n p + n = k | a n | r n p ) r p ( 1 + ( | ρ | k p + r k p ) n = k | a n | ) d k r p + ( B A ) r k d k ( B A ) | ρ | k p

and

|f(z)| r p ( a p n = k | a n | r n p ) = r p ( 1 + n = k ( | ρ | n p r n p ) | a n | ) .
(30)

If rρ, then we obtain |f(z)| r p . If r>ρ, then the sequence {( ρ n p r n p )} is decreasing and negative. Thus, by (30), we obtain

|f(z)| r p ( 1 ( r k p | ρ | k p ) n = 2 a n ) d k r p ( B A ) r k d k ( B A ) | ρ | k p

and we have the assertion (27). Making use of Lemma 2, in conjunction with (17), we readily obtain the assertion (29) of Theorem 4. □

Putting ρ=0 in Theorem 4 we have the following corollary.

Corollary 6 Let f TW 0 η , |z|=r<1. If d k d n (n N k ), then

r p B A d k r k |f(z)| r p + B A d k r k .

Moreover, if n d k k d n (n N k ), then

p r p 1 k ( B A ) d k r k 1 | f (z)|p r p 1 + k ( B A ) d k r k 1 .
(31)

The result is sharp, with the extremal function f k , η of the form (24).

4 The radii of convexity and starlikeness

Theorem 5 If p>0, then

R α ( T W η ) = inf n k ( ( p α ) d n ( n α ) ( B A ) ) 1 n p .
(32)

The functions f n , η of the form

f n , η (z)= a p ( z p B A d n e i ( p n ) η z n ) (zU;n N k ; a p >0)
(33)

are the extremal functions.

Proof A function f T η (p,k) of the form (1) is starlike of order α in U(r) if and only if it satisfies the condition (7). Since

| z f ( z ) f ( z ) p|=| n = k ( n p ) a n z n a p z p + n = k a n z n | n = k ( n p ) | a n | | z | n p a p n = k | a n | | z | n p ,

the condition (7) is true if

n = k n α p α | a n | r n p a p .
(34)

By Theorem 2, we have

n = k d n B A | a n | a p .
(35)

Thus, the condition (34) is true if

n α p α r n p d n B A (n N k ),

that is, if

r ( ( p α ) d n ( n α ) ( B A ) ) 1 n p (n N k ).
(36)

It follows that each function f TW η is starlike of order α in U(r), where

r= inf n k ( ( p α ) d n ( n α ) ( B A ) ) 1 n p .

The functions f n , η of the form (33) realize equality in (35), and the radius r cannot be larger. Thus we have (32). □

The following result may be proved in much the same way as Theorem 5.

Theorem 6 If p>0, then

R α c ( T W η ) = inf n k ( ( p α ) d n n ( n α ) ( B A ) ) 1 n p .

The functions f n , η of the form (33) are the extremal functions.

It is clear that for

a p = d n d n ( B A ) | ρ | n p >0

the extremal function f n , η of the form (33) belongs to the class TW ρ η . Moreover, we have

T W ρ η T W η .

Thus, by Theorems 5 and 6 we have the following corollary.

Corollary 7 Let the sequence { d n (BA)|ρ | n p } be positive, p>0. Then

R α ( T W ρ η ) = inf n k ( ( p α ) d n ( n α ) ( B A ) ) 1 n p , R α c ( T W ρ η ) = inf n k ( ( p α ) d n n ( n α ) ( B A ) ) 1 n p .

5 Subordination results

Before stating and proving our subordination theorems for the class TW η , we need the following definition and lemma.

Definition 1 A sequence { b n } of complex numbers is said to be a subordinating factor sequence if for each function f S c we have

n = 1 b n a n z n f(z)( a 1 =1).
(37)

Lemma 3 [36]

A sequence { b n } is a subordinating factor sequence if and only if

{ 1 + 2 n = 1 b n z n } >0(zD).
(38)

Theorem 7 Let the sequence { d n } satisfy the inequality (25). If g S c and f TW η , then

[ ε z 1 p f ( z ) ] g(z)g(z)
(39)

and

[ z 1 p f ( z ) ] > 1 2 ε (zD),
(40)

where

ε= d k 2 a p ( B A + d k ) .
(41)

If p and (kp) are odd, and η=0, then the constant factor ε cannot be replaced by a larger number.

Proof Let a function f of the form (1) belong to the class TW η and suppose that a function g of the form

g(z)= n = 1 c n z n ( c 1 =1;zD)

belongs to the class S c . Then

[ ε z 1 p f ( z ) ] g(z)= n = 1 b n c n z n (zD),

where

b n ={ ε a p if  n = 1 , 0 if  2 n k p , ε a n + p 1 if  n > k p .

Thus, by Definition 1, the subordination result (39) holds true if { b n } is the subordinating factor sequence. By (25), we have

{ 1 + 2 n = 1 b n z n } = { 1 + 2 ε a p z + n = k d k B A + d k a n z n p } 1 2 ε r r ( B A + d k ) a p n = k d n | a n | ( | z | = r < 1 ) .

Thus, by using Theorem 2, we obtain

{ 1 + 2 n = 1 b n z n } 1 d k B A + d k r B A B A + d k r>0.

This evidently proves the inequality (38) and hence the subordination result (39). The inequality (40) follows from (39) by taking

g(z)= z 1 z = n = 1 z n (zD).

Next, we observe that the function f k , η of the form (33) belongs to the class TW η . If p and (kp) are odd, and η=0, then

z 1 p f k , η (z) | z = 1 = 1 2 ε

and the constant (41) cannot be replaced by any larger one. □

Remark 1 By using (17) in Theorem 7, we obtain the result related to the class TW ρ η . Moreover, by putting ρ=0, we have the following corollary.

Corollary 8 Let the sequence { d n } satisfy the inequality (25). If g S c and f TW 0 η , then conditions (39) and (40) hold true. If p and (kp) are odd, and η=0, then the constant factor ε= d k 2 ( B A + d k ) cannot be replaced by a larger number.

6 Integral means inequalities

Due to Littlewood [22], we obtain integral means inequalities for the functions from the class TW η .

Lemma 4 [22]

Let f, g be functions analytic in U. If fg, then

0 2 π |f ( r e i θ ) | λ dθ 0 2 π |g ( r e i θ ) | λ dθ(0<r<1,0<λ).
(42)

Applying Lemma 4 and Theorem 2, we prove the following result.

Theorem 8 Let the sequence { d n } satisfy (25), k=p+1. If fT W ρ , then

0 2 π |f(z) | λ dθ 0 2 π | f p + 1 , η (z) | λ dθ ( 0 < r < 1 , 0 < λ ; z = r e i θ ) ,
(43)

where f p + 1 , η is defined by (33).

Proof For function f of the form (1), the inequality (43) is equivalent to the following:

0 2 π | a p + n = p + 1 a n z n p | λ dθ 0 2 π | a p B A d p + 1 e i η z | λ dθ ( z = r e i θ ) .

By Lemma 4, it suffices to show that

n = p + 1 a n z n p B A d p + 1 e i η z.
(44)

Setting

w(z)= n = p + 1 d p + 1 e i η B A a n z n p (zD)

and using (25) and Theorem 2 we obtain

|w(z)|=| n = p + 1 d p + 1 B A a n z n p ||z| n = p + 1 d n B A | a n ||z|(zD)

and

n = p + 1 a n z n p = B A d p + 1 e i η w(z)(zD).

Thus, by definition of subordination we have (44) and this completes the proof. □

By using (17) in Theorem 8 we have the following corollary.

Corollary 9 Let the sequence { d n } satisfy (25), k=p+1. If fT W ρ η , then

0 2 π |f ( r e i θ ) | λ dθ 0 2 π | f p + 1 , η ( r e i θ ) | λ dθ ( 0 < r < 1 , λ > 0 ; z = r e i θ ) ,

where f p + 1 , η is defined by (21).

7 Partial sums

Let f be a function of the form (1). Due to Silvia [27], we investigate the partial sums f m of the function f defined by

f k 1 (z)= a p z p , f m (z)= a p z p + n = k m a n z n (m N k ).
(45)

In this section, we consider partial sums of functions in the class TW η and obtain sharp lower bounds for the ratios of real part of f to f m and f to f m .

Theorem 9 Let the sequence { d n } be increasing and d k BA. If f TW η , then

Re { f ( z ) f m ( z ) } 1 B A d m + 1 (zD,m N k 1 )
(46)

and

Re { f m ( z ) f ( z ) } d m + 1 B A + d m + 1 (zD,m N k 1 ).
(47)

The bounds are sharp, with the extremal functions f m + 1 , η defined by (21).

Proof

Since

d n + 1 B A > d n B A >1(n N k ),

by Theorem 1, we have

n = k m | a n |+ d m + 1 B A n = m + 1 | a n | n = k d n B A | a n | a p .
(48)

Let

g(z)= d m + 1 B A { f ( z ) f m ( z ) ( 1 B A d m + 1 ) } =1+ d m + 1 B A n = m + 1 a n z n p a p + n = k m a n z n p (zD).
(49)

Applying (48), we find that

| g ( z ) 1 g ( z ) + 1 | d m + 1 B A n = m + 1 | a n | 2 a p 2 n = 2 n | a n | d m + 1 B A n = m + 1 | a n | 1(zD).

Thus, we have [g(z)]0 (zD) and by (49) we have the assertion (46) of Theorem 9. Similarly, if we take

h(z)= ( 1 + d m + 1 B A ) { f m ( z ) f ( z ) d m + 1 B A + d m + 1 } (zD)

and making use of (48), we can deduce that

| h ( z ) 1 h ( z ) + 1 | ( 1 + d m + 1 B A ) n = m + 1 | a n | 2 a p 2 n = k m | a n | ( d m + 1 B A 1 ) n = m + 1 | a n | 1(zD),

which leads us immediately to the assertion (47) of Theorem 9. In order to see that the function f m + 1 , η of the form (21) gives the results sharp, we observe that

f m + 1 , η ( z ) ( f m + 1 , η ) m ( z ) = 1 B A d m + 1 ( z = e i η ) , ( f m + 1 , η ) m ( z ) f m + 1 , η ( z ) = d m + 1 B A + d m + 1 ( z = e i ( η + π m p + 1 ) ) .

This completes the proof. □

Theorem 10 Let the sequence { d n } be increasing and d k >(m+1)(BA). If f TW η , then

Re { f ( z ) f m ( z ) } 1 ( m + 1 ) ( B A ) d m + 1 ( z D , m N k 1 ) , Re { f m ( z ) f ( z ) } d m + 1 ( m + 1 ) ( B A ) + d m + 1 ( z D , m N k 1 ) .

The bounds are sharp, with the extremal functions f m + 1 , η defined by (21).

Proof

By setting

g(z)= d m + 1 B A { f ( z ) f m ( z ) ( 1 ( m + 1 ) ( B A ) d m + 1 ) } (zD)

and

h(z)= ( m + 1 + d m + 1 B A ) { f m ( z ) f ( z ) d m + 1 ( m + 1 ) ( B A ) + d m + 1 } (zD),

the proof is analogous to that of Theorem 9, and we omit the details. □

Remark 2 By using (17) in Theorems 9 and 10, we obtain the results related to the class TW ρ η .

8 Concluding remarks

We conclude this paper by observing that, in view of the subordination relation (9), choosing the functions ϕ and φ, we can consider new and also well-known classes of functions. Let p>0, nN, x n =1 and

W ρ n (p,k;φ;A,B;δ):= W ρ ( p , k ; z φ ( z ) p , l = 0 n 1 φ ( x l z ) ; A , B ; δ ) .

The class W ρ n (p,k;φ;A,B;δ) generalize well-known classes, which were investigated in earlier works; see, for example, [5, 23, 28, 30]. In particular, the class W ρ n (p,k;φ;A,B;0) contains functions fM(p,k), which satisfies the condition

z ( φ f ) ( z ) l = 0 n 1 ( φ f ) ( x l z ) p 1 + A z 1 + B z .

It is related to the class of starlike functions with respect to n-symmetric points. Moreover, putting n=1, we obtain the class W ρ 1 (p,k;φ;A,B;0) defined by the following condition:

z ( φ f ) ( z ) ( φ f ) ( z ) p 1 + A z 1 + B z .

The class is related to the class of starlike functions. In particular, we have

S p (α):= W ρ 1 ( 1 , 2 ; z p 1 z ; 2 α 1 , 1 ; 0 ) .

Analogously, the class

W ρ n (p,k;φ;2γp,1;δ)(0γ<p)

contains functions fM(p,k), which satisfy the condition

{ z ( φ f ) ( z ) l = 0 n 1 ( φ f ) ( x l z ) γ } >δ| z ( φ f ) ( z ) l = 0 n 1 ( φ f ) ( x l z ) p|(zD).

It is related to the class of δ-uniformly convex functions of order γ with respect to n-symmetric points. Moreover, putting n=1, we obtain the class W n (p,k;φ;2γp,1;δ) defined by the following condition:

{ z ( φ f ) ( z ) ( φ f ) ( z ) γ } >δ| z ( φ f ) ( z ) ( φ f ) ( z ) p|(zD).

The class is related to the class of δ-uniformly convex functions of order γ. The classes

UST ( γ , δ ) : = W 0 ( 1 , 2 ; z 1 z ; 2 γ 1 , 1 ; δ ) , UCV ( γ , δ ) : = W 0 ( 1 , 2 ; z ( 1 z ) 2 ; 2 γ 1 , 1 ; δ ) ,

are the well-known classes of δ-starlike functions of order γ and δ-uniformly convex functions of order γ, respectively. In particular, the classes UCV:=UCV(1,0), δUCV:=UCV(δ,0) were introduced by Goodman [18], and Wisniowska et al. [29] and [19], respectively (see also [20]).

We note that the class

H T (φ;γ,δ):= T 0 (1,2) W n (1,2;φ;2γ1,1;δ)

was introduced and studied by Raina and Bansal [24].

If we set

h( α 1 ,z):= z q F s ( α 1 ,, α q ; β 1 ,, β s ;z),

where F s q is the generalized hypergeometric function, then we obtain the class

UH(q,s,λ,γ,δ):= H T ( λ h ( α 1 + 1 , z ) + ( 1 λ ) h ( α 1 , z ) ; γ , δ ) (0λ1)

defined by Srivastava et al. [26].

Let λ be a convex parameter. A function fM(p,k) belongs to the class

V λ (φ;A,B):=W ( λ φ ( z ) z + ( 1 λ ) φ ( z ) , z ; A , B ; 0 )

if it satisfies the following condition:

λ ( φ f ) ( z ) z +(1λ) ( φ f ) (z) 1 + A z 1 + B z .

Moreover, a function fM(p,k) belongs to the class

D λ (φ;A,B):=W ( λ φ ( z ) z + ( 1 λ ) φ ( z ) ; A , B ; 0 )

if it satisfies the following condition:

z ( φ f ) ( z ) + ( 1 λ ) z 2 ( φ f ) ( z ) λ ( φ f ) ( z ) + ( 1 λ ) z ( φ f ) ( z ) 1 + A z 1 + B z .

The considered classes are defined by using the convolution φf or equivalently by the linear operator

J φ :M(p,k)M(p,k), J φ (f)=φf.

By choosing the function φ, we can obtain a lot of important linear operators, and in consequence new and also well-known classes of functions. We can listed here some of these linear operators as the Salagean operator, the Cho-Kim-Srivastava operator, the Dziok-Raina operator, the Hohlov operator, the Dziok-Srivastava operator, the Carlson-Shaffer operator, the Ruscheweyh derivative operator, the generalized Bernardi-Libera-Livingston operator, the fractional derivative operator and so on (see, for the precise relationships [14, 17]).

If we apply the results presented in the paper to the classes discussed above, we can lead to several results. Some of these were obtained in earlier works; see, for example, [317, 21, 2326, 3035].

References

  1. Montel P: Leçons sur les Fonctions Univalentes ou Multivalentes. Gauthier-Villars, Paris; 1933.

    Google Scholar 

  2. Silverman H: Univalent functions with varying arguments. Houst. J. Math. 1981, 7: 283–287.

    Google Scholar 

  3. Srivastava HM, Owa S: Certain classes of analytic functions with varying arguments. J. Math. Anal. Appl. 1988, 136: 217–228. 10.1016/0022-247X(88)90127-8

    MathSciNet  Article  Google Scholar 

  4. Aouf MK, Dziok J: Distortion and convolutional theorems for operators of generalized fractional calculus involving Wright function. J. Appl. Anal. 2008, 14(2):183–192.

    MathSciNet  Article  Google Scholar 

  5. Aouf MK, Hossen HM, Srivastava HM: Some families of multivalent functions. Comput. Math. Appl. 2000, 39: 39–48.

    MathSciNet  Article  Google Scholar 

  6. Aouf MK, Srivastava HM: Some families of starlike functions with negative coefficients. J. Math. Anal. Appl. 1996, 203(3):762–790. 10.1006/jmaa.1996.0411

    MathSciNet  Article  Google Scholar 

  7. Chen M-P, Irmak H, Srivastava HM: Some families of multivalently analytic functions with negative coefficients. J. Math. Anal. Appl. 1997, 214(2):674–690. 10.1006/jmaa.1997.5615

    MathSciNet  Article  Google Scholar 

  8. Cho NE, Bulboaca T, Srivastava HM: A general family of integral operators and associated subordination and superordination properties of some special analytic function classes. Appl. Math. Comput. 2012, 219: 2278–2288. 10.1016/j.amc.2012.08.075

    MathSciNet  Article  Google Scholar 

  9. Cho NE, Kim IH, Srivastava HM: Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator. Appl. Math. Comput. 2010, 217: 918–928. 10.1016/j.amc.2010.06.036

    MathSciNet  Article  Google Scholar 

  10. Dziok J: A new class of multivalent analytic functions defined by the Hadamard product. Demonstr. Math. 2011, 44(2):233–251.

    MathSciNet  Google Scholar 

  11. Dziok J: Classes of functions defined by certain differential-integral operators. J. Comput. Appl. Math. 1999, 105(1–2):245–255. 10.1016/S0377-0427(99)00014-X

    MathSciNet  Article  Google Scholar 

  12. Dziok J: Applications of the Jack lemma. Acta Math. Hung. 2004, 105: 93–102.

    MathSciNet  Article  Google Scholar 

  13. Dziok J: On the convex combination of the Dziok-Srivastava operator. Appl. Math. Comput. 2007, 188(2):1214–1220. 10.1016/j.amc.2006.10.081

    MathSciNet  Article  Google Scholar 

  14. Dziok J, Raina RK: Families of analytic functions associated with the Wright generalized hypergeometric function. Demonstr. Math. 2004, 37(3):533–542.

    MathSciNet  Google Scholar 

  15. Dziok J, Raina RK, Srivastava HM: Some classes of analytic functions associated with operators on Hilbert space involving Wright’s generalized hypergeometric function. Proc. Jangieon Math. Soc. 2004, 7: 43–55.

    MathSciNet  Google Scholar 

  16. Dziok J, Srivastava HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 2003, 14: 7–18. 10.1080/10652460304543

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  18. Goodman AW: On uniformly convex functions. Ann. Pol. Math. 1991, 56: 87–92.

    Google Scholar 

  19. Kanas S, Wisniowska A: Conic regions and k -uniform convexity. J. Comput. Appl. Math. 1999, 105: 327–336. 10.1016/S0377-0427(99)00018-7

    MathSciNet  Article  Google Scholar 

  20. Kanas S, Srivastava HM: Linear operators associated with k -uniformly convex functions. Integral Transforms Spec. Funct. 2000, 9: 121–132. 10.1080/10652460008819249

    MathSciNet  Article  Google Scholar 

  21. Kulkarni SR, Naik UH, Srivastava HM: An application of fractional calculus to a new class of multivalent functions with negative coefficients. Comput. Math. Appl. 1999, 38(5–6):169–182. 10.1016/S0898-1221(99)00224-2

    MathSciNet  Article  Google Scholar 

  22. Littlewood JE: On inequalities in theory of functions. Proc. Lond. Math. Soc. 1925, 23: 481–519. 10.1112/plms/s2-23.1.481

    Article  Google Scholar 

  23. Liu J-L, Srivastava HM: Certain properties of the Dziok-Srivastava operator. Appl. Math. Comput. 2004, 159: 485–493. 10.1016/j.amc.2003.08.133

    MathSciNet  Article  Google Scholar 

  24. Raina RK, Bansal D: Some properties of a new class of analytic functions defined in terms of a Hadamard product. J. Inequal. Pure Appl. Math. 2008., 9: Article ID 22

    Google Scholar 

  25. Raina RK, Srivastava HM: A unified presentation of certain subclasses of prestarlike functions with negative coefficients. Comput. Math. Appl. 1999, 38(11–12):71–78. 10.1016/S0898-1221(99)00286-2

    MathSciNet  Article  Google Scholar 

  26. Ramachandran C, Shanmugam TN, Srivastava HM, Swaminathan A: A unified class of k -uniformly convex functions defined by the Dziok-Srivastava linear operator. Appl. Math. Comput. 2007, 190: 1627–1636. 10.1016/j.amc.2007.02.042

    MathSciNet  Article  Google Scholar 

  27. Silvia EM: Partial sums of convex functions of order α . Houst. J. Math. 1985, 11: 397–404.

    MathSciNet  Google Scholar 

  28. Sokół J: On some applications of the Dziok-Srivastava operator. Appl. Math. Comput. 2008, 201: 774–780. 10.1016/j.amc.2008.01.013

    MathSciNet  Article  Google Scholar 

  29. Sokół J, Wiśniowska A: On some classes of starlike functions related with parabola. Folia Sci. Univ. Tech. Resov. 1993, 121: 35–42.

    Google Scholar 

  30. Srivastava HM, Murugusundaramoorthy G, Sivasubramanian S: Hypergeometric functions in the parabolic starlike and uniformly convex domains. Integral Transforms Spec. Funct. 2007, 18: 511–520. 10.1080/10652460701391324

    MathSciNet  Article  Google Scholar 

  31. Wang Z-G, Li Q-G, Jiang Y-P: Certain subclasses of multivalent analytic functions involving the generalized Srivastava-Attiya operator. Integral Transforms Spec. Funct. 2010, 21: 221–234. 10.1080/10652460903098248

    MathSciNet  Article  Google Scholar 

  32. Wang Z-G, Liu Z-H, Xiang R-G: Some criteria for meromorphic multivalent starlike functions. Appl. Math. Comput. 2011, 218: 1107–1111. 10.1016/j.amc.2011.03.079

    MathSciNet  Article  Google Scholar 

  33. Wang Z-G, Liu Z-H, Cataş A: On neighborhoods and partial sums of certain meromorphic multivalent functions. Appl. Math. Lett. 2011, 24(6):864–868. 10.1016/j.aml.2010.12.033

    MathSciNet  Article  Google Scholar 

  34. Wang Z-G, Sun Y, Xu N: Some properties of certain meromorphic close-to-convex functions. Appl. Math. Lett. 2012, 25: 454–460. 10.1016/j.aml.2011.09.035

    MathSciNet  Article  Google Scholar 

  35. Wang Z-G, Sun Y, Zhang Z-H: Certain classes of meromorphic multivalent functions. Comput. Math. Appl. 2009, 58: 1408–1417. 10.1016/j.camwa.2009.07.020

    MathSciNet  Article  Google Scholar 

  36. Wilf HS: Subordinating factor sequence for convex maps of the unit circle. Proc. Am. Math. Soc. 1961, 12: 689–693. 10.1090/S0002-9939-1961-0125214-5

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jacek Dziok.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Dziok, J. Classes of multivalent analytic and meromorphic functions with two fixed points. Fixed Point Theory Appl 2013, 86 (2013). https://doi.org/10.1186/1687-1812-2013-86

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2013-86

Keywords

  • analytic functions
  • varying arguments
  • fixed points
  • Montel’s normalization
  • subordination
  • Hadamard product