MATEMATIQKI VESNIK 65, 3 (2013), 373 382 September 2013 originalni nauqni rad researh paper CERTAIN SUFFICIENT CONDITIONS FOR A SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR S. Sivasubramanian, Thomas Rosy and K. Muthunagai Abstrat. Making use of the Hohlov operator, we obtain inlusion relations between the lasses of ertain normalized analyti funtions. Relevant onnetions of our work with the earlier works are pointed out. 1. Introdution Let A be the lass of funtions f normalized by f = z + whih are analyti in the open unit disk a n z n, (1.1) U = {z : z C and z < 1}. As usual, we denote by S the sublass of A onsisting of funtions whih are also univalent in U. A funtion f A is said to be starlike of order α (0 α < 1), if and only if ( zf ) R > α (z U). f This funtion lass is denoted by S (α). We also write S (0) =: S, where S denotes the lass of funtions f A that are starlike in U with respet to the origin. A funtion f A is said to be onvex of order α (0 α < 1) if and only if ( R 1 + zf ) f > α (z U). 2010 Mathematis Subjet Classifiation: 33C45, 33A30, 30C45 Keywords and phrases: Starlike funtion; onvolution, negative oeffiients; oeffiient inequalities; growth and distortion theorems. 373
374 S. Sivasubramanian, T. Rosy, K. Muthunagai This lass is denoted by K(α). Further, K = K(0), the well-known standard lass of onvex funtions. It is an established fat that f K(α) zf S (α). Let M(λ, α) be a sublass of A onsisting of funtions of the form that satisfy the ondition ( z(zf ) ) R (1 λ)zf + λz(zf ) > α, z for some α and λ where 0 α < 1 and 0 λ < 1. That is, M(λ, α) be a sublass of S onsisting of funtions of the form that satisfy the ondition ( f + zf ) R f + λzf > α, z The lass M(λ, α) was introdued by Altintas and Owa [1] and also investigated very reently by Mostafa [12]. A funtion f A is said to be in the lass UCV of uniformly onvex funtions in U if and only if it has the property that, for every irular ar δ ontained in the unit disk U, with enter ζ also in U, the image urve f(δ) is a onvex ar. The funtion lass UCV was introdued by Goodman [7]. Furthermore, we denote by k UCV and k ST, (0 k < ), two interesting sublasses of S onsisting respetively of funtions whih are k-uniformly onvex and k-starlike in U. Namely, we have for 0 k < and k UCV := { ( f S : R 1 + zf ) f > k zf f }, (z U) { ( zf ) k ST := f S : R > k zf } f f 1, (z U). The lass k UCV was introdued by Kanas and Wiśniowska [10], where its geometri definition and onnetions with the oni domains were onsidered. The lass k ST was investigated in [11]. In fat, it is related to the lass k UCV by means of the well-known Alexander equivalene between the usual lasses of onvex and starlike funtions (see also the work of Kanas and Srivastava [9] for further developments involving eah of the lasses k UCV and k ST ). In partiular, when k = 1, we obtain k UCV UCV and k ST = SP, where UCV and SP are the familiar lasses of uniformly onvex funtions and paraboli starlike funtions in U respetively (see for details, [7]). Indeed, by making use of a ertain frational alulus operator, Srivastava and Mishra [17] presented a systemati and unified study of the lasses UCV and SP.
Let us denote (see [10,11]) 8(aros k) 2 P 1 (k) = Certain suffiient onditions... 375 for 0 k < 1 π 2 (1 k 2 ) 8 π 2 for k = 1 π 2 4 t(1 + t)(k 2 1)K 2 (t) for k > 1 (1.2) where t (0, 1) is determined by k = osh(πk (t)/[4k(t)]), K is the Legendre s omplete ellipti integral of the first kind K(t) = 1 0 dx (1 x2 )(1 t 2 x 2 ) and K (t) = K( 1 t 2 ) is the omplementary integral of K(t). Let Ω k be a domain suh that 1 Ω k and Ω k = { w = u + iv : u 2 = k 2 (u 1) 2 + k 2 v 2}, 0 k <. The domain Ω k is ellipti for k > 1, hyperboli when 0 < k < 1, paraboli when k = 1, and a right half-plane when k = 0. If p is an analyti funtion with p(0) = 1 whih maps the unit dis U onformally onto the region Ω k, then P 1 (k) = p (0). P 1 (k) is stritly dereasing funtion of the variable k and it values are inluded in the interval (0, 2]. Let f A be of the form (1.1). If f k UCV, then the following oeffiient inequalities hold true (f. [10]): a n (P 1(k)) n 1, n M \ {1}. (1.3) n! Similarly, if f of the form (1.1) belongs to the lass k ST, then (f. [11]) a n (P 1(k)) n 1, n M \ {1}. (n 1)! A funtion f A is said to be in the lass R τ (A, B), (τ C\{0}, 1 B < A 1), if it satisfies the inequality f 1 (A B)τ B[f 1] < 1 (z U). The lass R τ (A, B) was introdued earlier by Dixit and Pal [5]. Two of the many interesting sublasses of the lass R τ (A, B) are worthy of mention here. First of all, by setting τ = e iη os η ( π/2 < η < π/2), A = 1 2β (0 β < 1) and B = 1, the lass R τ (A, B) redues essentially to the lass R η (β) introdued and studied by Ponnusamy and Rønning [14], where R η (β) = { f A : R(e iη (f β)) > 0 (z U; π/2 < η < π/2, 0 β < 1) }.
376 S. Sivasubramanian, T. Rosy, K. Muthunagai Seondly, if we put τ = 1, A = β and B = β (0 < β 1), we obtain the lass of funtions f A satisfying the inequality f 1 f + 1 < β (z U; 0 < β 1) whih was studied by (among others) Padmanabhan [13] and Caplinger and Causey [3], (see also the works [6, 15, 16, 18]). The Gaussian hypergeometri funtion F (a, b; ; z) given by 2F 1 (a, b; ; z) = F (a, b; ; z) = n=0 (a) n (b) n () n (1) n z n (z U) is the solution of the homogenous hypergeometri differential equation z(1 z)w + [ (a + b + 1)z]w abw = 0 and has rih appliations in various fields suh as onformal mappings, quasi onformal theory, ontinued frations and so on. By the Gauss Summation Theorem, we get F (a, b; ; 1) = n=0 (a) n (b) n () n (1) n = Γ( a b)γ() Γ( a)γ( b) for R( a b) > 0. Here, a, b, are omplex numbers suh that 0, 1, 2, 3,..., (a) 0 = 1 for a 0, and for eah positive integer n, (a) n = a(a + 1)(a + 2)... (a + n 1) is the Pohhammer symbol. In the ase of = k, k = 0, 1, 2,..., F (a, b; ; z) is defined if a = j or b = j where j k. In this situation, F (a, b; ; z) beomes a polynomial of degree j with respet to z. Results regarding F (a, b; ; z) when R( a b) is positive, zero or negative are abundant in the literature. In partiular when R( a b) > 0, the funtion is bounded. The hypergeometri funtion F (a, b; ; z) has been studied extensively by various authors and play an important role in Geometri Funtion Theory. It is useful in unifying various funtions by giving appropriate values to the parameters a, b and. We refer to [4, 6, 15, 16] and referenes therein for some important results. For funtions f A given by (1.1) and g A given by g = z + b nz n, we define the Hadamard produt (or onvolution) of f and g by (f g) = z + a n b n z n, z U. For f A, we reall the operator I a,b, (f) of Hohlov [8] whih maps A into itself defined by means of Hadamard produt as I a,b, (f) = zf (a, b; ; z) f.
Certain suffiient onditions... 377 Therefore, for a funtion f defined by (1.1), we have I a,b, (f) = z + Using the integral representation, F (a, b; ; z) = we an write [I a,b, (f)] = Γ() Γ(b)Γ( b) 1 0 Γ() Γ(b)Γ( b) () n 1 (1) n 1 a n z n. t b 1 (1 t) b 1 dt, R() > R(b) > 0, (1 tz) a 1 0 t b 1 b 1 f(tz) z (1 t) dt t (1 tz) a. When f equals the onvex funtion z 1 z, then the operator I a,b,(f) in this ase beomes zf (a, b; ; z). If a = 1, b = 1 + δ, = 2 + δ with R(δ) > 1 then the onvolution operator I a,b, (f) turns into Bernardi operator B f = [I a,b, (f)] = 1 + δ z δ 1 0 t δ 1 f(t) dt. Indeed, I 1,1,2 (f) and I 1,2,3 (f) are known as Alexander and Libera operators, respetively. To prove the main results, we need the following Lemmas. Lemma 1. [1] A funtion f A belongs to the lass M(λ, α) if n(n λαn α + λα) a n 1 α. (1.4) Lemma 2. [5] If f R τ (A, B) is of form (1.1) then The result is sharp. a n (A B) τ, n M \ {1}. (1.5) n In this paper, we estimate ertain inlusion relations involving the lasses k UCV, k ST and M(λ, α). 2. Main results In this paper, we will study the ation of the hypergeometri funtion on the lasses k UCV, k ST. Theorem 1. Let a, b C \ {0}. Also, let be a real number suh that > a + b + 1. If f R τ (A, B), and if the inequality Γ()Γ( a b 1) Γ( a )Γ( b ) is satisfied, then I a, b, (f) M(λ, α). [(1 λα) ab + (1 α)( a b 1)] ( ) 1 (1 α) (A B) τ + 1 (2.1)
378 S. Sivasubramanian, T. Rosy, K. Muthunagai Proof. Let f be of the form (1.1) belong to the lass R τ (A, B). By virtue of Lemma 1, it suffies to show that n(n λαn α + λα) a n () n 1 (1) n 1 1 α. Taking into aount the inequality (1.5) and the relation (a) n 1 ( a ) n 1, we dedue that (n λαn α + λα) a n () n 1 (1) n 1 (A B) τ (1 λα) n () n 1 (1) n 1 + (A B) τ α(λ 1) ( a ) n 1 ( b ) n 1 () n 1 (1) n (A B) τ {(1 λα) { = (A B) τ (1 λα) ab ( a ) n 1 ( b ) n 1 () n 1 (1) n 2 + (1 α) F (1 + a, 1 + b, 1 + ; 1) } + (1 α) (F ( a, b, ; 1) 1) where we use the relation ( a ) n 1 ( b ) } n 1 () n 1 (1) n 1 (a) n = a(a + 1) n 1. (2.2) The proof now follows by an appliation of Gauss summation theorem and (2.1). For the hoie of b = a, we have the following orollary. Corollary 1. Let a C \ {0}. Also, let be a real number suh that > 2 a + 1. If f R τ (A, B), and if the inequality Γ()Γ( 2 a 1) (Γ( a )) 2 [ (1 λα) a 2 + (1 α)( 2 a 1) ] is satisfied, then I a, a, (f) M(λ, α). ( ) 1 (1 α) (A B) τ + 1 In the speial ase when b = 1, Theorem 1 immediately yields a result onerning the Carlson-Shaffer operator L(a, )(f) := I a, 1, (f) (see [4]). Corollary 2. Let a C \ {0}. Also, let be a real number suh that > a + 1. If f R τ (A, B), and if the inequality ( ) Γ()Γ( a 2) 1 [(1 λα) a + (1 α)( a 2)] (1 α) Γ( a )Γ( 1) (A B) τ + 1 is satisfied, then L(a, )(f) M(λ, α).
Certain suffiient onditions... 379 Theorem 2. Let a, b C \ {0}. Also, let be a real number and P 1 = P 1 (k) be given by (1.2). If, for some k (0 k < ), f k UCV, and the inequality (1 λα) ab P 1 3F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 2; 1) + α(λ 1) 3 F 2 ( a, b, P 1 ;, 1; 1) 1 αλ (2.3) is satisfied, then I a, b, (f) M(λ, α). Proof. Let f be given by (1.1). By (1.4), to show I a, b, (f) M(λ, α), it is suffiient to prove that n(n λαn α + λα) a n () n 1 (1) n 1 1 α. We will repeat the method of proving used in the proof of Theorem 1. Applying the estimates for the oeffiients given by (1.3), and making use of the relations (2.2) and (a) n ( a ) n, we get n(n λαn α + λα) a n () n 1 (1) n 1 [n(1 λα) + α(λ 1)] ( a ) n 1( b ) n 1 (P 1 ) n 1 () n 1 (1) n 1 (1) n 1 = (1 λα) ab P 1 + α(λ 1) = (1 λα) ab P 1 1 α, 3 (1 + a ) n 2 (1 + b ) n 2 (1 + P 1 ) n 2 (1 + ) n 2 (1) n 2 (2) n 2 () n 1 (1) n 1 (1) n 1 F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 2; 1) + α(λ 1) [ 3 F 2 ( a, b, P 1 ;, 1; 1) 1] provided the ondition (2.3) is satisfied. For the hoies of b = a and b = 1, we an dedue further orollaries of Theorem 2 as follows. Corollary 3. Let a C \ {0}. Suppose that b = a. Also, let be a real number and P 1 = P 1 (k) be given by (1.2). If f k UCV for some k (0 k < ) and the inequality (1 λα) a 2 P 1 3F 2 (1 + a, 1 + a, 1 + P 1 ; 1 +, 2; 1) + α(λ 1) 3 F 2 ( a, a, P 1 ;, 1; 1) 1 αλ is satisfied, then I a, a, (f) M(λ, α).
380 S. Sivasubramanian, T. Rosy, K. Muthunagai Corollary 4. Let a C \ {0}. Also, let be a real number and P 1 = P 1 (k) be given by (1.2). If f k UCV for some k (0 k < ) and the inequality (1 λα) a P 1 3F 2 (1 + a, 2, 1 + P 1 ; 1 +, 2; 1) + α(λ 1) 3 F 2 ( a, 1, P 1 ;, 1; 1) 1 αλ is satisfied, then L(a, )(f) M(λ, α). Theorem 3. Let a, b C \ {0}. Also, let be a real number and P 1 = P 1 (k) be given by (1.2). If f k ST, for some k (0 k < ), and the inequality (1 λα) ab P 1 3F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 1; 1) + (2 α λα) ab P 1 3F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 2; 1) + (1 α) 3 F 2 ( a, b, P 1 ;, 1; 1) 2(1 α) (2.4) is satisfied, then I a, b, (f) M(λ, α). Proof. Let f be given by (1.1). We will repeat the method of proving used in the proof of Theorem 1. Applying the estimates for the oeffiients given by (1.4), and making use of the relations (2.2) and (a) n ( a ) n, we get n(n λαn α + λα) a n () n 1 (1) n 1 n[n(1 λα) + α(λ 1)] ( a ) n 1( b ) n 1 (P 1 ) n 1 () n 1 (1) n 1 (1) n 1 = (n 1)[(n 1)(1 λα) + (1 α)] ( a ) n 1( b ) n 1 (P 1 ) n 1 () n 1 (1) n 1 (1) n 1 + [(n 1)(1 λα) + (1 α)] ( a ) n 1( b ) n 1 (P 1 ) n 1 () n 1 (1) n 1 (1) n 1 = (1 λα) + (1 α) () n 1 (1) n 2 (1) n 2 () n 1 (1) n 1 (1) n 2 + (1 λα) = (1 λα) = (1 λα) ab P 1 () n 1 (1) n 1 (1) n 2 () n 1 (1) n 2 (1) n 2 () n 1 (1) n 1 (1) n 2 + (1 α) + (1 α) + (2 α λα) 3F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 1; 1) () n 1 (1) n 1 (1) n 1 () n 1 (1) n 1 (1) n 1 + (2 α λα) ab P 1 3F 2 (1 + a, 1 + b, 1 + P 1 ; 1 +, 2; 1) + (1 α) [ 3 F 2 ( a, b, P 1 ;, 1; 1) 1] 1 α provided the ondition (2.4) is satisfied.
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