Partial Sums of Normalized Wright Functions
aa r X i v : . [ m a t h . C V ] J un PARTIAL SUMS OF NORMALIZED WRIGHT FUNCTIONS
MUHEY U DIN , MOHSAN RAZA , NIHAT YA ˘GMUR ∗ Abstract.
In this paper we find the partial sums of two kinds normalizedWright functions and the partial sums of Alexander transform of these nor-malized Wright functions. Introduction and preliminaries
Let A be the class of functions f of the form f ( z ) = z + ∞ X m =2 a m z m analytic in the open unit disc U = { z : | z | < } . Consider the Alexander trans-form given as: A [ f ] ( z ) = z Z f ( t ) t dt = z + ∞ X m =2 a m m z m . The surprize use of Hypergeometric function in the solution of the Bieberbachconjecture has attracted many researchers to study the special functions. Manyauthors who study on geometric functions theory are intersted in some geometricproperties such as univalency, starlikeness, convexity and close-to-convexity ofspecial functions. Recently, several researchers have studied the geometric prop-erties of hypergeometric functions [12, 28], Bessel functions [1, 2, 3, 4, 5, 6, 22,23, 24], Struve functions [14, 30], Lommel functions [8]. Motivated by the aboveworks Prajpat [19] studied some geometric properties of Wright function W λ,µ ( z ) = ∞ X m =0 z m m !Γ ( λm + µ ) , λ > − , µ ∈ C . This series is absolutely convergent in C , when λ > − U for λ = − . Furthermore this function is entire. The Wrightfunctions were introduced by Wright [29] and have been used in the asymtotictheory of partitions, in the theory of integral transforms of the Hankel type andin Mikusinski operational calculus. Recently, Wright functions have been foundin the solution of partial differential equations of fractional order. It was foundthat the corresponding Green functions can be represented in terms of the Wrightfunctions [18, 21] . For positive rational number λ, the Wright functions can be Date : ∗ Corresponding author2010
Mathematics Subject Classification.
Key words and phrases.
Partial sums, Analytic functions, Normalized Wright functions. represented in terms of generalized hypergeometric functions. For some detailssee [10, section 2.1]. In particular, the functions W ,v +1 ( − z /
4) can be expressedin terms of the Bessel functions J v , given as: J v ( z ) = (cid:16) z (cid:17) W ,v +1 ( − z /
4) = ∞ X m =0 ( − m ( z/ m + v m !Γ ( m + v + 1) . The Wright function generalizes various functions like Array functions, Whittakerfunctions, entire auxiliary functions, etc. For the details, we refer to [10] . Prajapatdiscussed some geometric properties of the following normalizations of Wrightfunctions in [19] W λ,µ ( z ) = Γ ( µ ) zW λ,µ ( z )= z + ∞ X m =1 Γ ( µ ) m !Γ ( λm + µ ) z m +1 . λ > − , µ > , z ∈ U , (1.1) W λ,µ ( z ) = Γ ( λ + µ ) (cid:20) W λ,µ ( z ) −
1Γ ( µ ) (cid:21) = z + ∞ X m =1 Γ ( λ + µ )( m + 1)!Γ ( λm + λ + µ ) z m +1 , z ∈ U , (1.2)where λ > − , λ + µ > . The Pochhammer (or Appell) symbol, defined in termsof Euler’s gamma functions is given as ( x ) n = Γ( x + n ) / Γ( x ) = x ( x +1) ... ( x + n − .
1) and (1 . W λ,µ ) n ( z ) = z + n X m =1 Γ( µ ) m !Γ( λm + µ ) z m +1 when thecoefficients of W λ,µ satisfy certain conditions. We determine the lower bounds ofRe (cid:26) W λ,µ ( z ) ( W λ,µ ) n ( z ) (cid:27) , Re (cid:26) ( W λ,µ ) n ( z ) W λ,µ ( z ) (cid:27) , Re (cid:26) W ′ λ,µ ( z ) ( W λ,µ ) ′ n ( z ) (cid:27) , Re (cid:26) ( W λ,µ ) ′ n ( z ) W ′ λ,µ ( z ) (cid:27) , Re (cid:26) A [ W λ,µ ] ( z ) ( A [ W λ,µ ]) n ( z ) (cid:27) , Re (cid:26) ( A [ W λ,µ ]) n ( z ) A [ W λ,µ ] ( z ) (cid:27) , where A [ W λ,µ ] is the Alexander trans-form of W λ,µ . Some similar results are obtained for the function W λ,µ ( z ) . Forsome works on partial sums, we refer [7, 11, 13, 15, 17, 25, 26, 27].
Lemma 1.1.
Let λ, µ ∈ R and λ > − , µ > . Then the function W λ,µ : U → C defined by (1 . satisfies the following inequalities:(i) If µ > , then |W λ,µ ( z ) | ≤ µ + 12 µ − , z ∈ U . (ii) If µ > , then (cid:12)(cid:12) W ′ λ,µ ( z ) (cid:12)(cid:12) ≤ µ + 1 µ − , z ∈ U . (iii) If µ > , then | A [ W λ,µ ] ( z ) | ≤ µ µ − , z ∈ U . RIGHT FUNTIONS 3
Proof. (i) By using the well-known triangle inequalitiy | z + z | ≤ | z | + | z | with the inequality Γ ( µ + m ) ≤ Γ ( µ + mλ ) , m ∈ N , which is equivalent to Γ( µ )Γ( λm + µ ) ≤ µ ( µ +1) ... ( µ + m − = µ ) m , m ∈ N and the inequalities( µ ) m ≥ µ m , m ! ≥ m − , m ∈ N , we obtain |W λ,µ ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + ∞ X m =1 Γ ( µ ) m !Γ ( λm + µ ) z m +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X m =1 Γ ( µ ) m !Γ ( λm + µ ) ≤ ∞ X m =1 m ! ( µ ) m ≤ µ ∞ X m =1 (cid:18) µ (cid:19) m − = 2 µ + 12 µ − , µ > / , z ∈ U .(ii) To prove (ii), we use the well-known triangle inequality with the inequality Γ( µ )Γ( λm + µ ) ≤ µ ( µ +1) ... ( µ + m − = µ ) m , m ∈ N and the inequalities( µ ) m ≥ µ m , m ! ≥ m + 12 , m ∈ N , we have (cid:12)(cid:12) W ′ λ,µ ( z ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X m =1 Γ ( µ ) ( m + 1) m !Γ ( λm + µ ) z m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X m =1 Γ ( µ ) ( m + 1) m !Γ ( λm + µ ) ≤ ∞ X m =1 m + 1 m ! ( µ ) m ≤ µ ∞ X m =1 (cid:18) µ (cid:19) m − = µ + 1 µ − , µ > , z ∈ U .(iii) Making the use of triangle inequality with Γ( µ )Γ( λm + µ ) ≤ µ ) m and the inequalities( µ ) m ≥ µ m , ( m + 1)! ≥ m , m ∈ N , M. U. DIN, M. RAZA, N. YA ˘GMUR we have | A [ W λ,µ ] ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + ∞ X m =1 Γ ( µ )( m + 1)!Γ ( λm + µ ) z m +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X m =1 Γ ( µ )( m + 1)!Γ ( λm + µ ) ≤ ∞ X m =1 m + 1)! ( µ ) m ≤ µ ∞ X m =1 (cid:18) µ (cid:19) m − = 2 µ µ − , µ > / , z ∈ U . (cid:3) Lemma 1.2.
Let λ, µ ∈ R and λ > − , λ + µ > . Then the function W λ,µ : U → C defined by (1 . satisfies the following inequalities:(i) If λ + µ > , then | W λ,µ ( z ) | ≤ λ + µ )2 ( λ + µ ) − , z ∈ U . (ii) If λ + µ > , then (cid:12)(cid:12) W ′ λ,µ ( z ) (cid:12)(cid:12) ≤ λ + µ ) + 12 ( λ + µ ) − , z ∈ U . Proof. (i) By using the well-known triangle inequality | z + z | ≤ | z | + | z | with the inequality Γ ( λ + µ + m ) ≤ Γ ( mλ + λ + µ ) , m ∈ N , which is equivalentto Γ( λ + µ )Γ( mλ + λ + µ ) ≤ λ + µ )( λ + µ +1) ... ( λ + µ + m − = λ + µ ) m , m ∈ N and the inequalities( λ + µ ) m ≥ ( λ + µ ) m , m ! ≥ m − , m ∈ N , we obtain | W λ,µ ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + ∞ X m =1 Γ ( λ + µ ) m !Γ ( λm + λ + µ ) z m +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X m =1 Γ ( λ + µ ) m !Γ ( λm + λ + µ ) ≤ ∞ X m =1 m ! ( λ + µ ) m ≤ λ + µ ∞ X m =1 (cid:18)
12 ( λ + µ ) (cid:19) m − = 2 ( λ + µ ) + 12 ( λ + µ ) − , λ + µ ) > / , z ∈ U . RIGHT FUNTIONS 5 (ii) By using the well-known triangle inequality with the inequality Γ( λ + µ )Γ( mλ + λ + µ ) ≤ λ + µ )( λ + µ +1) ... ( λ + µ + m − = λ + µ ) m , m ∈ N and the inequalities( λ + µ ) m ≥ ( λ + µ ) m , m ! ≥ m + 12 , m ∈ N , we have (cid:12)(cid:12) W ′ λ,µ ( z ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X m =1 Γ ( λ + µ ) ( m + 1) m !Γ ( λm + λ + µ ) z m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X m =1 Γ ( λ + µ ) ( m + 1) m !Γ ( λm + λ + µ ) ≤ ∞ X m =1 m + 1 m ! ( λ + µ ) m ≤ λ + µ ) ∞ X m =1 (cid:18) λ + µ (cid:19) m − = ( λ + µ ) + 1( λ + µ ) − , ( λ + µ ) > , z ∈ U . (cid:3) Partial Sums of W λ,µ ( z ) Theorem 2.1.
Let λ, µ ∈ R such that λ > − , µ > . Then Re (cid:26) W λ,µ ( z )( W λ,µ ) n ( z ) (cid:27) ≥ µ − µ − , z ∈ U . (2.1) and Re (cid:26) ( W λ,µ ) n ( z ) W λ,µ ( z ) (cid:27) ≥ µ − µ + 1 , z ∈ U . (2.2) Proof.
By using (i) of Lemma 1 .
1, it is clear that1 + ∞ X m =1 | a m | ≤ µ + 12 µ − , which is equivalent to 2 µ − ∞ X m =1 | a m | ≤ . M. U. DIN, M. RAZA, N. YA ˘GMUR where a m = Γ( µ ) m !Γ( λm + µ ) . Now, we may write2 µ − (cid:26) W λ,µ ( z )( W λ,µ ) n ( z ) − µ − µ − (cid:27) = 1 + n X m =1 a m z m + (cid:0) µ − (cid:1) ∞ X m = n +1 a m z m n X m =1 a m z m = : 1 + w ( z )1 − w ( z ) . Then it is clear that w ( z ) = (cid:0) µ − (cid:1) ∞ X m = n +1 a m z m n X m =1 a m z m + (cid:0) µ − (cid:1) ∞ X m = n +1 a m z m and | w ( z ) | ≤ (cid:0) µ − (cid:1) ∞ X m = n +1 | a m | − n X m =1 | a m | − (cid:0) µ − (cid:1) ∞ X m = n +1 | a m | . This implies that | w ( z ) | ≤ (cid:18) µ − (cid:19) ∞ X m = n +1 | a m | ≤ − n X m =1 | a m | . Which further implies that n X m =1 | a m | + (cid:18) µ − (cid:19) ∞ X m = n +1 | a m | ≤ . (2.3)It suffices to show that the left hand side of (2 .
3) is bounded above by (cid:0) µ − (cid:1) ∞ X m =1 | a m | , which is equivalent to 2 µ − n X m =1 | a m | ≥ . RIGHT FUNTIONS 7
To prove (2 . , we write2 µ + 12 (cid:26) ( W λ,µ ) n ( z ) W λ,µ ( z ) − µ − µ + 1 (cid:27) = 1 + n X m =1 a m z m − (cid:0) µ − (cid:1) ∞ X m = n +1 a m z m ∞ X m =1 a m z m = 1 + w ( z )1 − w ( z ) . Therefore | w ( z ) | ≤ (cid:0) µ +12 (cid:1) ∞ X m = n +1 | a m | − n X m =1 | a m | − (cid:0) µ − (cid:1) ∞ X m = n +1 | a m | ≤ . The last inequality is equivalent to n X m =1 | a m | + (cid:18) µ − (cid:19) ∞ X m = n +1 | a m | ≤ . (2.4)Since the left hand side of (2 .
4) is bounded above by (cid:0) µ − (cid:1) ∞ X m =1 | a m | , this com-pletes the proof. (cid:3) Theorem 2.2.
Let λ, µ ∈ R , with λ > − and µ > . Then Re (cid:26) W ′ λ,µ ( z )( W λ,µ ) ′ n ( z ) (cid:27) ≥ µ − µ − , z ∈ U . (2.5)Re ( ( W λ,µ ) ′ n ( z ) W ′ λ,µ ( z ) ) ≥ µ − µ + 1 , z ∈ U . (2.6) Proof.
From part (ii) of Lemma 1.1, we observe that1 + ∞ X m =1 ( m + 1) | a m | ≤ µ + 1 µ − , where a m = Γ( µ ) m !Γ( λm + µ ) . This implies that (cid:18) µ − (cid:19) ∞ X m =1 ( m + 1) | a m | ≤ . M. U. DIN, M. RAZA, N. YA ˘GMUR
Consider (cid:18) µ − (cid:19) (cid:26) W ′ λ,µ ( z )( W λ,µ ) ′ n ( z ) − µ − µ − (cid:27) = 1 + n X m =1 ( m + 1) a m z m + (cid:0) µ − (cid:1) ∞ X m = n +1 ( m + 1) a m z m n X m =1 ( m + 1) a m z m = 1 + w ( z )1 − w ( z ) . Therefore | w ( z ) | ≤ (cid:0) µ − (cid:1) ∞ X m = n +1 ( m + 1) | a m | − n X m =1 ( m + 1) | a m | − (cid:0) µ − (cid:1) ∞ X m = n +1 ( m + 1) | a m | ≤ . The last inequality is equivalent to n X m =1 ( m + 1) | a m | + (cid:18) µ − (cid:19) ∞ X m = n +1 ( m + 1) | a m | ≤ . (2.7)It suffices to show that the left hand side of (2 .
7) is bounded above by (cid:0) µ − (cid:1) ∞ X m =1 | a m | ( m + 1) . Which is equivalent to µ − n X m =1 ( m + 1) | a m | ≥ . To prove the result (2 . , we write (cid:18) µ + 12 (cid:19) ( ( W λ,µ ) ′ n ( z ) W ′ λ,µ ( z ) − µ − µ + 1 ) = 1 + w ( z )1 − w ( z ) . Therefore | w ( z ) | ≤ (cid:0) µ +12 (cid:1) ∞ X m = n +1 ( m + 1) | a m | − n X m =1 ( m + 1) | a m | − µ − ∞ X m = n +1 ( m + 1) | a m | ≤ . The last inequality is equivalent to n X m =1 | a m | ( m + 1) + µ − ∞ X m = n +1 ( m + 1) | a m | ≤ . (2.8)It suffices to show that the left hand side of (2 .
8) is bounded above by
RIGHT FUNTIONS 9 µ − ∞ X m =1 ( m + 1) | a m | , the proof is complete. (cid:3) Theorem 2.3.
Let λ, µ ∈ R , with λ > − and µ > . Then Re (cid:26) A [ W λ,µ ] ( z )( A [ W λ,µ ]) n ( z ) (cid:27) ≥ µ − µ − , z ∈ U , (2.9) and Re (cid:26) ( A [ W λ,µ ]) n ( z ) A [ W λ,µ ] ( z ) (cid:27) ≥ µ − µ , z ∈ U , (2.10) where A [ W λ,µ ] is the Alexander transform of W λ,µ . Proof.
To prove (2 . , we consider from part (iii) of Lemma 1.1 so that1 + ∞ X m =1 | a m | ( m + 1) ≤ µ µ − , which is equvalent to (2 µ − ∞ X m =1 | a m | ( m + 1) ≤ , where a m = Γ( µ ) m !Γ( λm + µ ) . Now, we write(2 µ − (cid:26) A [ W λ,µ ] ( z )( A [ W λ,µ ]) n ( z ) − µ − µ − (cid:27) = 1 + n X m =1 a m ( m +1) z m + (2 µ − ∞ X m = n +1 a m ( m +1) z m n X m =1 a m ( m +1) z m = 1 + w ( z )1 − w ( z ) , where | w ( z ) | ≤ (2 µ − ∞ X m = n +1 | a m | ( m +1) − n X m =1 | a m | ( m +1) − (2 µ − ∞ X m = n +1 | a m | ( m +1) ≤ . The last inequality is equivalent to n X m =1 | a m | ( m + 1) + (2 µ − ∞ X m = n +1 | a m | ( m + 1) ≤ . (2.11)It suffices to show that the left hand side of (2 .
11) is bounded above by(2 µ − ∞ X m =1 | a m | ( m +1) , which is equivalent to (2 µ − ∞ X m =1 | a m | ( m +1) ≥ . This completesthe proof.
The proof of (2 .
10) is similar to the proof of Theorem 2.1. (cid:3)
Remark 2.4.
For λ = 1 , µ = 5 / W , / ( − z ) = (cid:16) sin(2 √ z )2 √ z − cos(2 √ z ) (cid:17) , and for n = 0 , we have (cid:0) W , / (cid:1) ( z ) = z, so,Re (cid:18) sin(2 √ z ) − √ z cos(2 √ z )2 z √ z (cid:19) ≥
23 ( z ∈ U ) , (2.12)and Re (cid:18) z √ z sin(2 √ z ) − √ z cos(2 √ z ) (cid:19) ≥
12 ( z ∈ U ) . (2.13)The image domains of f ( z ) = sin(2 √ z ) − √ z cos(2 √ z )2 z √ z and g ( z ) = z √ z sin(2 √ z ) − √ z cos(2 √ z ) are shown in Figure 1. 3. Partial Sums of W λ,µ ( z ) Theorem 3.1.
Let λ, µ ∈ R , with λ > − and µ + λ > . Then Re (cid:26) W λ,µ ( z )( W λ,µ ) n ( z ) (cid:27) ≥ λ + µ ) −
22 ( λ + µ ) − , z ∈ U , (3.1) and Re (cid:26) ( W λ,µ ) n ( z ) W λ,µ ( z ) (cid:27) ≥ λ + µ ) −
12 ( λ + µ ) , z ∈ U , (3.2) where W λ,µ ( z ) is the normalized Wright function.Proof. By using Lemma 1 . ∞ X m =1 | a m | ≤ λ + µ )2 ( λ + µ ) − , where a m = Γ( λ + µ )( m +1)!Γ( λm + λ + µ ) . This implies that { λ + µ ) − } ∞ X m =1 | a m | ≤ . RIGHT FUNTIONS 11
Now we may write { λ + µ ) − } (cid:26) W λ,µ ( z )( W λ,µ ) n ( z ) − λ + µ ) −
22 ( λ + µ ) − (cid:27) = 1 + n X m =1 a m z m + { λ + µ ) − } ∞ X m = n +1 a m z m n X m =1 a m z m = 1 + w ( z )1 − w ( z ) . It is clear that w ( z ) = { λ + µ ) − } ∞ X m = n +1 a m z m n X m =1 a m z m + { λ + µ ) − } ∞ X m = n +1 a m z m , and | w ( z ) | ≤ { λ + µ ) − } ∞ X m = n +1 | a m | − n X m =1 | a m | − { λ + µ ) − } ∞ X m = n +1 | a m | . This implies that | w ( z ) | ≤ n X m =1 | a m | + { λ + µ ) − } ∞ X m = n +1 | a m | ≤ . (3.3)It suffices to show that the left hand side of (3 .
3) is bounded above by { λ + µ ) − } ∞ X m =1 | a m | , which is equivalent to { λ + µ ) − } ∞ X m =1 | a m | ≥ . To prove (3 . , we consider that2 ( λ + µ ) (cid:26) ( W λ,µ ) n ( z ) W λ,µ ( z ) − λ + µ ) −
12 ( λ + µ ) (cid:27) . = 1 + n X m =1 a m z m + { λ + µ ) − } ∞ X m = n +1 a m z m ∞ X m =1 a m z m = 1 + w ( z )1 − w ( z ) . Therefore | w ( z ) | ≤ { λ + µ ) } ∞ X m = n +1 | a m | − n X m =1 | a m | − { λ + µ ) − } ∞ X m = n +1 | a m | . The last inequality is equivalent to n X m =1 | a m | + { λ + µ ) − } ∞ X m = n +1 | a m | ≤ . (3.4)Since the left hand side of (3 .
4) is bounded above by { λ + µ ) − } ∞ X m =1 | a m | , theproof is complete. (cid:3) Similarly, we have the following result.
Theorem 3.2.
Let λ, µ ∈ R , with λ > − and µ + λ > . Then Re (cid:26) W ′ λ,µ ( z )( W λ,µ ) ′ n ( z ) (cid:27) ≥ λ + µ ) −
32 ( λ + µ ) − , z ∈ U , (3.5) and Re ( ( W λ,µ ) ′ n ( z ) W ′ λ,µ ( z ) ) ≥ λ + µ ) −
12 ( λ + µ ) + 1 , z ∈ U , (3.6) where W λ,µ ( z ) is the normalized Wright function.Proof. Proof is similar to the Theorem 2.2. (cid:3)
Recently Ravichandran [20] presented a survey article on geometric propertiesof partial sums of univalent functions. Using Noshiro-Warschawski Theorem [9]for n = 0 in the inequalities (2 .
5) of Theorem 2.2 and (3 .
5) of Theorem 3.2, thefunctions W λ,µ ( z ) and W λ,µ ( z ) are univalent and also close to convex. Noshiro[16] showed that the radius of starlikness of f n ( the partial sums of the function f ∈ A ) is 1 /M if f satisfies the inequality | f ′ ( z ) | ≤ M. This implies that byusing the parts (ii) of Lemma 1.1 and Lemma 2.1, the radii of starlikeness of thefunctions ( W λ,µ ) n ( z ) and ( W λ,µ ) n ( z ) are µ − µ +1 and λ + µ ) − λ + µ )+1 respectively. Acknowledgement:
The research of N. Ya˘gmur is supported by ErzincanUniversity Rectorship under ”The Scientific and Research Project of ErzincanUniversity”, Project No: FEN-A-240215-0126.
References [1] ´A. Baricz, Functional inequalities involving special functions, J. Math. Anal. Appl., 319(2006), 450-459.[2] ´A. Baricz, Functional inequalities involving special functions. II, J. Math. Anal. Appl., 327(2007), 1202-1213.[3] ´A. Baricz, Generalized Bessel Functions of the First Kind, Lecture Notes in Mathemat-icsVol. 1994, Springer-Verlag, Berlin, 2010.
RIGHT FUNTIONS 13 [4] ´A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen, 73(2008), 155-178.[5] ´A. Baricz, Some inequalities involving generalized Bessel functions, Math. Inequal. Appl.,10 (2007), 827-842.[6] ´A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions,Integral Transforms Spec. Funct., 21,9 (2010), 641-653.[7] L. Brickman, D. J. Hallenbeck, T. H. Macgregor and D. Wilken. Convex hulls and extreme-points of families of starlike and convex mappings. Trans. Amer. Math. Soc. 185 (1973),413–428.[8] M. C¸ a˘glar and E. Deniz, Partial sums of the normalized Lommel functions, Math. Inequal.Appl. 18, 3 (2015), 1189-1199.[9] A. W. Goodman, Univalent functions, Vol 1. Marinar Publi. Comp., Tempa Florida, 1984.[10] R. Gorenflo, Y. Luchko and F. Mainardi. Analytic properties and applications of Wrightfunctions. Frac. Cal. App. Anal. 2, 4 (1999), 383-414.[11] L.J. Lin and S. Owa. On partial sums of the Libera integral operator. J. Math. Anal. Appl.213, 2 (1997), 444–454.[12] S. S. Miller, P. T Mocanu, Univalence of Gaussian and confluent hypergeometric functions,Proc Amer Math Soc., 110, 2 (1990), 333–342.[13] H. Orhan and E. Gunes. Neighborhoods and partial sums of analytic functions based onGaussian hypergeometric functions. Indian J. Math. 51, 3 (2009), 489–510.[14] H. Orhan and N. Ya˘gmur, Geometric properties of generalized Struve functions, An. Stiint.Univ. Al. I. Cuza Iasi. Mat. (N.S.) (doi: 10.2478/aicu-2014-0007).[15] H. Orhan and N. Ya˘gmur. Partial sums of generalized Bessel functions. J. Math.Ineq. 8, 4(2014), 863-877.[16] K. Noshiro, On the starshaped mapping by analytic function. Proc. Imp. Acad. 8, 7 (1932),275-277.[17] S. Owa, H. M. Srivastava and N. Saito. Partial sums of certain classes of analytic functions.Int. J. Comput. Math. 81, 10 (2004), 1239–1256.[18] I. Podlubny, Fractional differential equations. San Diego: Academic press, 1999.[19] J. K. Prajapat. Certain geometric properties of the Wright function. Integ. Trans. SpecFunc. 26, 3 (2015), 203-212.[20] V. Ravichandran, Geometric properties of partial sums of univalent functions. arXiv:1207.4302v1.[21] S. G. Samko, A. A. Kilbas and O. I. Marichev. Fractional integrals and derivatives: theoryand applications. Gordan and Breach: New York, 1933.[22] V. Selinger, Geometric properties of normalized Bessel functions, Pure Math. Appl. 6(1995), 273–277.[23] R. Sz´asz, About the starlikeness of Bessel functions, Integral Transforms Spec. Funct. 25,9 (2014), 750-755.[24] R. Sz´asz and P. Kup´an, About the univalence of the Bessel functions, Stud. Univ. Babe¸s-Bolyai Math. 54(1) (2009), 127–132.[25] T. Sheil-Small. A note on partial sums of convex schlicht functions. Bull. London Math.Soc. 2 (1970), 165–168.[26] H. Silverman. Partial sums of starlike and convex functions. J. Math. Anal. Appl. 209(1997), 221–227.[27] E. M. Silvia. On partial sums of convex functions of order α. Houston J. Math. 11 (1985),397–404.[28] V. Singh St, Ruscheweyh, On the order of starlikeness of hypergeometric functions, J MathAnal Appl., 113(1986), 1–11.[29] E. M. Wright. On the coefficients of power series having exponential singularities. J. LondonMath. Soc. 8 (1933),71-79.[30] N. Ya˘gmur and H. Orhan. Partial sums of generalized Struve functions. Miskolc Math.Notes (accepted). Department of Mathematics, Government College University Faisalabad,Pakistan
E-mail address : [email protected]. Department of Mathematics, Government College University Faisalabad,Pakistan.
E-mail address : [email protected] Department of Mathematics, Erzincan University, Erzincan 24000, Turkey
E-mail address ::