STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION

Let q_(λ)(z)=1+λsinh(ζ),0<λ<1/sinh(1)be a non-vanishing analytic function in the open unit disk.We introduce a subclass S^(*)(q_(λ))of starlike functions which contains the functions f such that zf'/f is subordinated by q_(λ).We establish inclusion and radii results for the class S^(*)(q_(λ))for several known classes of starlike functions.Furthermore,we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class S^(*)(q_(λ)).We also find a sharp bound for the third Hankel determinant for the caseλ=1/2.

Differences in root stress and strain distribution in buccal and lingual orthodontics: A finite element analysis study

Orthodontic forces can cause stress and strain concentration and microcracks on tooth root surfaces.This study aimed to analyze whether a lingual orthodontic appliance was more likely to cause root stress concentration and root resorption than the traditional buccal appliance.A finite element model of the root of the maxillary central incisor with straight,buccal,and lingual curvatures was established.A load perpendicular to the tooth surface on the buccal and lingual surfaces of the central incisor was applied,and the stress and strain concentration at the root apex was compared.The stress and strain at the root apex of the lingual cingulum group were lesser than those of the buccal group;little difference was observed between the lingual and buccal clinical crown central augmentation groups.The stress and strain at the root apex of the lingual direction movement group were greater than those of the buccal direction movement group.The direction of the root curvature also influenced the amount of stress and strain at the root apex.The difference in the risk of root resorption between lingual and buccal orthodontics is dependent on the height of the bracket placement.The loading direction of the orthodontic force and direction of the root curvature also affect the stress and strain at the root apex.