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Abstract
We derive sufficient conditions for two integral operators to be convex (univalent) provided that the factors
are π½-spirallike
Results & Lemmas (7)
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Lemma 1.
Lemma 1. π(π§) is π½-spirallike if and only if there is a starlike function π (π§),such that π(π§) π§ = (π (π§) π§) πΌ (π§βπ), Where πΌ= πβππ½cosπ½.
Lemma 1. π(π§) is π½-spirallike if and only if there is a starlike function π (π§),such that π(π§) π§ = (π (π§) π§) πΌ (π§βπ), Where πΌ= πβππ½cosπ½.
Theorem 1.
Theorem 1. Let πΉπ(π§) be the integral operator given by (1), for πβ 1,β
β
β
β
, π, if (i) ππ is π½-spirallike for every π,(β π 2 < π½< π 2). (ii)ββ¦
Theorem 1. Let πΉπ(π§) be the integral operator given by (1), for πβ{1,β
β
β
β
, π}, if (i) ππ is π½-spirallike for every π,(β π 2 < π½< π 2). (ii)β 1 cosπ½π π π=1 < π+ 1. (iii) πΌπ= πππ½π. Then πΉπ(π§) is convex.
Corollary 1.
Corollary 1. Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if (i)πΌ= πππ½. (ii)cosπ½> π π+1,(π is a positive integer). Then: β«π‘ π((βπΌ)2+βπΌβ1 βπΌ )β¦
Corollary 1. Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if (i)πΌ= πππ½. (ii)cosπ½> π π+1,(π is a positive integer). Then: β«π‘ π((βπΌ)2+βπΌβ1 βπΌ ) (π(π‘)
Corollary 2
Corollary 2 Let π(π§) be π½-spirallike,(β π 6 < π½< π 6), if πΌ= πππ½ Then: β«π‘ (βπΌ)2+βπΌβ1 βπΌ (π(π‘) π‘) πΌ ππ‘ π§ 0
Corollary 2 Let π(π§) be π½-spirallike,(β π 6 < π½< π 6), if πΌ= πππ½ Then: β«π‘ (βπΌ)2+βπΌβ1 βπΌ (π(π‘) π‘) πΌ ππ‘ π§ 0
Theorem 2.
Theorem 2. Let πΊπ(π§) be the integral operator given by (2), for πβ 1,β
β
β
β
, π, if (i) ππ is π½-spirallike for every π,(β π 2 < π½< π 2). (ii)ββ¦
Theorem 2. Let πΊπ(π§) be the integral operator given by (2), for πβ{1,β
β
β
β
, π}, if (i) ππ is π½-spirallike for every π,(β π 2 < π½< π 2). (ii)β cosπ½π π π=1 < 1. (iii) πΌπ= πππ½π. Then πΊπ(π§) is convex.
Corollary 3.
Corollary 3. Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if (i)πΌ= πππ½. (ii)cosπ½< 1 π,(π is a positive integer). Then: β«(π(π‘) π‘) ππΌ ππ‘ π§
Corollary 3. Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if (i)πΌ= πππ½. (ii)cosπ½< 1 π,(π is a positive integer). Then: β«(π(π‘) π‘) ππΌ ππ‘ π§
Corollary 4
Corollary 4 Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if πΌ= πππ½ Then: β«(π(π‘) π‘) πΌ ππ‘ π§ 0
Corollary 4 Let π(π§) be π½-spirallike,(β π 2 < π½< π 2), if πΌ= πππ½ Then: β«(π(π‘) π‘) πΌ ππ‘ π§ 0
Definitions (3)
Def 1.
Definition 1. A function πβπ is called π½-spirallike if β(πππ½π§πβ²(π§) π(π§) ) > 0, (π§βπ) for a real number π½(β π 2 < π½< π 2). Also we denote byβ¦
Definition 1. A function πβπ is called π½-spirallike if β(πππ½π§πβ²(π§) π(π§) ) > 0, (π§βπ) for a real number π½(β π 2 < π½< π 2). Also we denote by πΎ the class of convex functions in π΄ normalized by π(0) = πβ²(0) β1 = 0, and β{π§πβ²β²(π§)
Def 2.
Definition 2. Let πΉπ(π§) = β«π‘ (βπΌ1)2+βπΌ1β1 βπΌ1 (π1(π‘) π‘ ) πΌ1 β
β
β
β
β
π‘ (βπΌπ)2+βπΌπβ1
Definition 2. Let πΉπ(π§) = β«π‘ (βπΌ1)2+βπΌ1β1 βπΌ1 (π1(π‘) π‘ ) πΌ1 β
β
β
β
β
π‘ (βπΌπ)2+βπΌπβ1
Def 3.
Definition 3. Let πΊπ(π§) = β«(π1(π‘) π‘ ) πΌ1 β
β
β
β
β
(ππ(π‘) π‘ ) πΌπ ππ‘
Definition 3. Let πΊπ(π§) = β«(π1(π‘) π‘ ) πΌ1 β
β
β
β
β
(ππ(π‘) π‘ ) πΌπ ππ‘
Function classes studied:
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